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UNIT Kinematics oVerAll eXpectAtions analyze technologies that apply concepts related to kinematics and assess the technologies social and environmental impact investigate in qualitative and quantitative terms linear motion with uniform and non-uniform velocity and solve related problems demonstrate an understanding of linear motion with uniform and non-uniform velocity BiG iDeAs Motion involves a change in the position of an object over time Motion can be described using mathematical relationships Many technologies that apply concepts related to kinematics have societal and environmental implications UNiT TASK PrEvIEw The challenge in this Unit Task is to design and construct a bean bag launcher You will need to calibrate the launcher to fire accurately at various target distances You will compete with your classmates to construct the most accurate launcher The Unit Task is described in detail on page As you work through the unit look for Unit Task Bookmarks to see how information in the section relates to the Unit Task Unit Kinematics NEL ---- SPLITTER -- Focus on STSE sports in Motion The interactions among Science Technology Society and the Environment STSE make physics relevant to our lives in a million different ways Sports are just one example In Canada hockey is more than just a game For many it s an obsession Hockey is an exciting fast-paced sport You can watch players skate down the ice to score the next goal a defenceman skillfully deflecting the puck out of the opposition player s control or a goaltender making a difficult glove save Speed is a critical part of the game from racing to get to the puck to firing a shot past the goaltender Imagine hockey or any other sport without motion and speed it would not be nearly as entertaining The rapid acceleration of the puck during a slapshot the way that a skilled player can rapidly change his or her speed and direction of motion these highspeed actions are what make a hockey game so exciting In other sports motion and speed are just as important for the athlete as for the enjoyment of the fans You can clearly see the skill of professional athletes in the precision control of a long soccer pass or of a basketball as it sails through the air in a perfect jumpshot There is a direct link between how objects move and the level of excitement we experience while watching or playing our favourite sports Questions Consider your favourite sport a What kinds of motions are required to play this sport Describe these motions in your own words b Describe the type s of motion that must be avoided to be successful in your favourite sport a List any advances in technology that have helped to make professionals in your favourite sport more successful b How have these advances in technology helped to improve the athlete s speed or motion Explain your reasoning c Research one advance in technology that has helped to make athletes in your favourite sport more successful Write a paragraph describing how this technology works How can a better understanding of motion help a participant in your favourite sport avoid injury What type of protective equipment is required in your favourite sport Is there any equipment that might help to make your favourite sport safer How does this equipment affect an athlete s motion Research how the use of protective equipment in your favourite sport has changed throughout its history Discuss your findings with a partner NEL Focus on STSE ---- SPLITTER -- unit ArE you rEAdy skills plotting a line graph on a Cartesian coordinate system analyzing graphs using and converting SI units solving an algebraic equation for an unknown variable using trigonometry to solve right triangles using a protractor and a centimetre ruler precisely effectively using a scientific calculator and a spreadsheet researching and collecting information planning and conducting investigations communicating scientific information clearly and accurately concepts motion Cartesian coordinate system Pythagorean theorem slope of a straight line Concepts review Recall the last time you rode in a car Describe the different types of motion that the vehicle underwent throughout the trip C An elevator is initially at rest on the second floor of a building A person on the tenth floor pushes the down button for the elevator Describe the motion of the elevator as it moves from the second floor to the tenth floor C What units are used to describe the following K u a distance b time c speed You are at the park watching a younger sibling swing back and forth on the swing set K u a Describe how you could measure the distance travelled and the time taken as your sibling swings back and forth b What sources of uncertainty exist in this experiment c How could you modify your experiment to reduce the uncertainty a Draw a Cartesian coordinate system Mark and label the compass directions north south east and west on your diagram Then mark and label the directions northwest and southeast on your diagram b Draw a line cm long starting at the origin of your Cartesian coordinate system and pointing northwest T I A tin can is placed outside just as it starts to rain Table contains measurements of the mass of water in the can taken at time intervals of s Describe the information provided in Table T I Table Mass Time Rainfall Data t s m g Skills review a Plot a graph of mass versus time using the data in Table Plot time on the horizontal axis b Determine an equation to represent the data Then answer the following questions i Describe the graph What is the relationship between mass and time ii Write an equation to determine the slope of the line on the graph and provide a value for the slope including the correct units iii What information does the slope provide iv Would it be valid to extrapolate this graph for another s v Using the equation of the graph determine the mass of water in the can at s T I C A Unit Kinematics NEL ---- SPLITTER -- Copy and complete Table Table Solve for a t T b t a D vf J J T I Equation F ma v d a d t Answer PV nRT y mx b d vit AB vf vi bt CD E a A robin flies a distance of cm How far has it flown in kilometres b What is the speed in metres per second of a car that is travelling at km h c What is the speed in kilometres per hour of a m s baseball pitch d How many seconds are there in a calendar year given that a calendar year has days in it T I Determine each unknown length in Figure a a a m b m c e f at v v ad f i c m Use the Pythagorean theorem to determine the length of the unknown side in each right triangle shown in Figure T I b b Figure d x m m z The three lines on the distance time graph in Figure represent the motion of three objects T I a Which object has travelled farthest at time t s b How far has each object travelled at time t s c What is the slope of each line object object object y b b Figure Ontario Physics U Pass m Approved U -F -OP USB FN Not Approved CrowleArt Group CO m Deborah Crowle th pass Pass d m a a m c c Ontario m Physics U FN CO U -F -OP USB CrowleArt Group Deborah Crowle rd pass Figure t s CAREER PATHWAYS PrevIew Throughout this unit you will see Career Links in the margins These links mention careers that are relevant to Kinematics On the Chapter Summary page at the end of each chapter you will find a Career Pathways feature that shows you the educational requirements of the careers There are also some career-related questions for you to research Approved Determine the value of each angle in the triangles in Ontario Physics U T I Question Not Approved a U -F -OP USB b FN CrowleArt Group c CO Deborah Crowle th pass Pass NEL Approved Not Approved Are You Ready ---- SPLITTER -- cHApter Motion in a Straight Line what Effects do Moving objects Have on Society and the Environment keY concepts After completing this chapter you will be able to explain how distance position and displacement are different acceleration are different are different explain how speed velocity and explain how vectors and scalars add and subtract vectors using scale diagrams and algebraic methods obtain motion information from position time velocity time and acceleration time graphs uniform acceleration problems using algebraic methods due to gravity affects the motion of objects close to the surface of Earth solve uniform velocity and describe how the acceleration assess the impact on society and the environment of a technology that applies concepts related to kinematics Automobiles have been made in North America for over years As technology has advanced automobile designs have changed substantially For example a Ford Model T could travel at a maximum speed of approximately km h This was considered a frightening speed at the time Over time vehicles have become faster Today many cars can reach speeds of km h or more much higher than the speed limits on any Canadian roads Scientists and engineers continue to develop a deeper understanding of motion and the factors that affect it This knowledge coupled with technological advances has enabled them to produce extremely fast experimental land vehicles The ThrustSSC SuperSonic Car reached an astounding speed of km h This extraordinary British-built vehicle was driven by a Royal Air Force pilot and powered by two jet engines The team that built the Thrust SSC and other competing groups are now attempting to build even faster vehicles Although we benefit greatly from motor vehicles that transport huge amounts of goods daily and make travel much easier we now realize that burning large amounts of fossil fuels has a negative impact on our environment Fuel consumption for the ThrustSSC was a mind-boggling L of gasoline per kilometre In comparison the passenger vehicles you see on the street consume about L km Many researchers have turned their attention to producing practical vehicles with a lower environmental impact Students at the University of Waterloo are moving the technology of passenger vehicles away from fossil fuels altogether They have designed and constructed a series of solar vehicles called Midnight Sun which have reached speeds of km h The team captured the world record for the longest journey by a solar-powered car Faster more efficient vehicles are an important part of the future of transportation in Canada Today s students are tomorrow s scientists and engineers With a sound understanding of the physics of motion we can improve today s transportation and environmental technologies to help protect our planet STARTiNg PoInTS Answer the following questions using your current knowledge You will have a chance to revisit these questions later applying concepts and skills from the chapter Give three examples of scientific language that can be used to describe the motion of objects Explain how graphs can provide information about the motion of an object How have the motion capabilities of automobiles changed over the past century How have changes in technologies that apply concepts about motion affected society and the environment Chapter Motion in a Straight Line NEL ---- SPLITTER -- C -P -OP USB Mini Investigation The Effect of Gravity on the Motion of objects Skills Predicting Performing Observing Analyzing SKILLS HANDBOOK A In this activity you will investigate how mass and shape change the effect of gravity on the motion of objects falling through the air Equipment and Materials spherical objects of different mass sheet of paper Pick up the two spheres and identify which is heavier Predict which sphere will reach the floor first if you release both of them simultaneously from the same height Record your prediction and explain your reasoning Hold the two spheres at arm s length from your body at the same height Release the two spheres Record your observations Use caution when dropping the spheres Do not drop them near your feet or near other students Pick up the spheres from the floor immediately Repeat Step with one sphere and a flat sheet of paper Record your observations Crumple the piece of paper used in Step into a ball of approximately the same size as one of the spheres Repeat Step with the crumpled paper ball and the sphere Record your observations A Did your observations in Step support the prediction you made in Step If not provide reasons why the prediction was not supported T I B Did the mass of each sphere in Step affect the time it took the spheres to reach the floor T I C Compare and contrast your observations for Step with those for Step T I D Compare and contrast your observations for Step with those for Step T I NEL Introduction ---- SPLITTER -- Distance position and Displacement You see and interact with moving objects every day Whether you are racing down a ski hill or running for a school bus motion is part of your everyday life Long jump athletes are very aware of distance position and displacement Long jumpers run down a stretch of track to a foul line and then jump as far as possible into a sand pit Figure Their goal is to increase the distance of their jumps To do this they focus on their speed strength and technique Successful long jumpers master this goal by applying the physics of motion C -P -OP USB LEARNiNg TIP SI Metric Units The SI system le Syst me international d unit s provides base units for measuring fundamental physical quantities Examples of base units are metres m for distance and displacement and seconds s for time Some physical quantities use derived units that are defined in terms of a combination of base units An example of a derived unit is metres per second m s which is the SI unit for speed and velocity Figure Long jumpers attempt to maximize the horizontal distance of their jumps describing the Motion of objects kinematics the study of motion motion a change in an object s location as measured by a particular observer distance d the total length of the path travelled by an object in motion direction the line an object moves along from a particular starting point CAREER LInK Many branches of engineering use principles of kinematics To learn more about becoming an engineer g o T o N E L So N S C i E N C E To understand the motion of objects we must first be able to describe motion Physicists use a number of specific terms and units to describe motion You are likely familiar with many of these terms and units Kinematics is the term used by physicists and engineers to describe the study of how objects move What exactly is motion Motion is a change in the location of an object as measured by an observer Distance in physics terms means the total length of the path travelled by an object in motion The SI metric base unit for distance is the metre m To help you understand the terms that describe motion imagine that you are at your home in Figure You are at the location marked m If you walk directly from home to your school in a straight line you will travel a distance of m If you walk from your school to the library and then return home you will travel an additional distance of m m m If your friend wants to know how to get to the library from your home telling him to walk for m is not very helpful You also need to tell your friend which direction to go Direction is the line an object moves along from a particular starting point expressed in degrees on a compass or in terms of the compass points north west east and south Directions can also be expressed as up down left right forward and backwards Directions are often expressed in brackets after the distance or other value For example m E indicates that the object is m to the east NEL Chapter Motion in a Straight Line ---- SPLITTER -- Direction is important when describing motion If the school in Figure is your starting point the library is in a different direction from your school than your home is If the library is your starting point then your school and home are in the same direction W home E school library m m m Figure Distance and direction along a straight line Scalar and vector Quantities A scalar quantity is a quantity that has magnitude size only Distance is an example of a scalar quantity Since direction is so important in describing motion physicists frequently use terms that include direction in their definitions A vector is a quantity that has magnitude size and also direction An arrow is placed above the symbol for a variable when it represents a vector quantity scalar a quantity that has only magnitude size vector a quantity that has magnitude size and direction Position and displacement Position is the distance and direction of an object from a particular reference point Position is a vector quantity represented by the symbol d Notice the vector arrow above the symbol d This arrow indicates that position is a vector it has a direction as well as a magnitude For example if home is your reference point the position of the school in Figure is m E Note that the magnitude of the position is the same as the straight-line distance m from home to school but the position also includes the direction due east E The position of the school from point m can be described by the equation d school m E WEB LInK To review scalar and vector quantities go To NELSoN SCiENCE position d the distance and direction of an object from a reference point Now assume that the library is your reference point or the point m The position of the school from the reference point library can be described by the equation d school m W Once the position of an object has been described you can describe what happens to the object when it moves from that position This is displacement the change in an object s position Displacement is represented by the symbol Dd Notice the vector arrow indicating that displacement is a vector quantity The tri- angle symbol is the Greek letter delta Delta is always read as change in so Dd is read as change in position As with any change displacement can be calculated by subtracting the initial position vector from the final position vector Dd d final d initial displacement Dd the change in position of an object When an object changes its position more than once experiences two or more displacements the total displacement Dd T of the object can be calculated by adding the displacements using the following equation Dd T Dd Dd NEL Distance Position and Displacement ---- SPLITTER -- Tutorial Calculating Displacement for Motion in a Straight Line When you walk from one place to another your position changes This change in your position is displacement The displacement can be calculated using your position at the beginning and the end of your journey with the equation Dd d final d initial Remember that position is a vector quantity so you have to take direction into account In the following Sample Problems we will calculate displacements using a range of techniques Refer to Figure for the first three Sample Problems In Sample Problem we will calculate the displacement of an object with an initial position of m mall W home m E school library m Figure m m Sample Problem Calculating Displacement from a Zero Starting Point by Vector Subtraction Imagine that you walk from home to school in a straight-line route What is your displacement Required Dd Analysis Dd d school d home Solution Dd d school d home m E m Dd m E Solution Figure shows that home is the starting point for your journey When you are at home your position has not changed Therefore your initial position is zero Your school has a position of m E relative to your home Given d school m E d home m Statement Your displacement when walking from your home to school is m E Sample Problem Calculating Displacement by Vector Subtraction Statement Your displacement when walking from school to the library is m E Defining the initial starting position of your motion as m will often make displacement problems simpler In Sample Problem if we had defined m as being the location of the school it would have been obvious from the diagram that the displacement from the school to the library is m E What is your displacement if you walk from your school to the library Note that all positions are measured relative to your home Given d school m E d library m E Required Dd Analysis Dd d library d school Solution Dd d library d school m E m E Dd m E Sample Problem Calculating Total Displacement by Vector Addition One night after working at the library you decide to go to the mall What is your total displacement when walking from the library to the mall Given Dd m W Dd m W from Figure Required Dd T If the displacement from the library to your home is represented by Dd and the displacement from your home to the mall is represented by Dd then the total displacement Dd T is given by vector addition of these two displacements Dd T Dd Dd Solution Dd T Dd Dd m W m W Dd T m W Analysis In this problem we are not simply calculating a change in position we are finding the sum of two different displacements The displacements are given in Figure To calculate the total displacement we will need to use vector addition This is simple if both vectors have the same direction Statement When walking from the library to the mall you experience a displacement of m W Chapter Motion in a Straight Line NEL ---- SPLITTER -- Sample Problem Calculating Total Displacement by Adding Displacements in Opposite Directions A dog is practising for her agility competition She leaves her trainer and runs m due west to pick up a ball She then carries the ball m due east and drops it into a bucket What is the dog s total displacement At this point it appears that we have a problem We need to add a vector with a direction W to a vector with a direction E We can transform this problem so that both vectors point in the same direction To do so consider the direction E to be the same as negative W The vector m E is the same as m W We can therefore rewrite the equation as follows m W m W Dd T m W In this problem the given values are displacements To calculate the total displacement we will add these two displacement vectors Given Dd m W Dd m E Required Dd T Analysis Dd T Dd Dd Solution Dd T Dd d Solution Practice m W m E Statement The dog s total displacement is m W A golfer hits a ball from a golf tee at a position of m W relative to the clubhouse The ball comes to rest at a position of m W relative to the clubhouse Determine the displacement of the golf ball T I ans m W A rabbit runs m N and stops to nibble on some grass The rabbit then hops m N to scratch against a small tree What is the rabbit s total displacement T I ans m N A skateboarder slides m up a ramp stops and then slides m down the ramp before jumping off What is his total displacement up the ramp T I ans m up vector Scale diagrams In Tutorial you used algebra to determine the displacement of an object in a straight line However there is another method you can use to solve displacement problems vector scale diagrams Vector scale diagrams show the vectors associated with a displacement drawn to a particular scale A vector can be represented by a directed line segment which is a straight line between two points with a specific direction Line segments have magnitude only Figure a A directed line segment is a line segment with an arrowhead pointing in a particular direction Figure b For example AB is a line segment in the direction from point A to point B Line segment BA is the same line segment but in the direction from point B to point A Figure c A directed line segment that represents a vector always has two ends The end with the arrowhead is referred to as the tip The other end is the tail A vector scale diagram is a representation of motion using directed line segments drawn to scale with arrowheads to show their specific directions Vector scale diagrams are very useful in measuring the total displacement of an object from its original position C -F -OP USB ai vector scale diagram a vector diagram drawn using a specific scale directed line segment a straight line between two points with a specific direction CAREER LInK Geomatics technicians collect data using GPS surveying and remote sensing They also record data using geographic information systems GIS GIS analysts capture and manage spatial information for use in government and industry To learn more about becoming a geomatics technician or GIS analyst g o T o N E L SoN S C i E N C E B B tip B tail c Figure a A line segment b directed line segment AB c directed line segment BA a b A tail A tip A NEL Distance Position and Displacement ---- SPLITTER -- Consider two displacements Dd m W and Dd m W We can determine the total displacement that results from adding these vectors together by drawing a vector scale diagram In general when drawing a vector scale diagram you should choose scales that produce diagrams approximately one-half to one full page in size The larger the diagram the more precise your results will be Figure shows a vector diagram drawn to a scale where cm in the diagram represents m in the real world Note that each vector in Figure has a tip the end with an arrowhead and a tail the other end Vectors can be added by joining them tip to tail This is similar to using a number line in mathematics Thus after applying our chosen scale Figure shows Dd drawn as a vector cm in length pointing due west The tip of Dd is joined to the tail of Dd In other words the displacement Dd is drawn as a directed line segment that is cm long pointing due west starting where the displacement Dd ends The total displacement Dd T is the displacement from the tail or start of the first vector to the tip or end of the second vector In this case Dd T points due west and has a length of cm Converting this measurement by applying our scale gives a total displacement of m W For straight-line motion vector scale diagrams are not very complex We will look at more advanced vector scale diagrams in Chapter when we consider motion in two dimensions N C -F -OP USB scale cm m dT m W d m W tail Figure Vector scale diagram d m W tip Tutorial Determining Total Displacement for Two Motions in Opposite Directions Using Vector Scale Diagrams In the following Sample Problem we will determine displacement by using vector scale diagrams Consider an example in which motion occurs in two opposite directions Sample Problem Using a Vector Scale Diagram to Determine the Total Displacement for Two Motions in Opposite Directions Imagine that you are going to visit your friend Before you get there you decide to stop at the variety store If you walk m N from your home to the store and then travel m S to your friend s house what is your total displacement C -F -OP USB N scale cm m store d m N home d dT m S Solution Let your initial displacement from your home to the store be Dd and your displacement from the store to your friend s house be Dd Ontario Physics U m N Dd Given Dd m S Required Dd T C -F -OP USB FN CrowleArt Group Analysis Dd TCO Dd Dd Deborah Crowle Solution Figure shows the given vectors with the tip of Dd th pass Pass joined to the tail of Dd The resultant vector Dd T is drawn in red Approved from the tail of Dd to the tip of Dd The direction of Dd T is S Dd T measures cm in length in Figure so using the scale of Not Approved cm m the actual magnitude of Dd T is m Statement Relative to your starting point at your home your total displacement is m S Chapter Motion in a Straight Line friend s house Figure Solution scale diagram for adding vectors with a change in direction Art is drawn to scale NEL ---- SPLITTER -- Practice A car drives m W to a stop sign It then continues on for a displacement of m W Use a vector scale diagram to determine the car s total displacement T I C ans m W A robin flies m S to catch a worm and then flies m N back to its nest Use a vector scale diagram to determine the robin s total displacement T I C ans m N Summary Motion involves a change in the position of an object Motion can be described using mathematical relationships A scalar is a quantity that has magnitude size only A vector is a quantity that has magnitude size and direction You can determine the displacement of an object by subtracting the start position from the end position You can determine total displacement by adding two or more displacements together algebraically or by using a vector scale diagram Vectors can be added by joining them tip to tail Questions Which of the following quantities are vectors and which are scalars Be sure to explain the reasoning for your answer K u C a A bird flies a distance of m b A train is travelling at km h due north c It takes an athlete s to run m Explain the following in your own words K u C a the difference between position and displacement b the difference between distance and displacement What is the displacement of a locomotive that changes its position from m W to m W T I A car changes its position from km W to km E What is the car s displacement T I Determine the total displacement for each of the following motions by algebraic methods and by using scale diagrams T I C a Dd m W Dd m W b Dd m W Dd m E c Dd m N Dd m S d Dd km W Dd km E Dd km W A person walks paces forward followed by paces forward and finally paces backwards T I C a Draw a vector scale diagram representing this person s motion Use a scale of cm pace b Check your answer by pacing out this motion yourself How close is your experimental result to that predicted by your vector scale diagram NEL Distance Position and Displacement ---- SPLITTER -- C -P -OP USB Speed and Velocity If you have been a passenger in a car or are taking driving lessons speed is something you have thought about Knowing the speed at which a vehicle is moving is important for safety Excessive speed is a contributing factor in many collisions Speed can be measured in different ways Police use laser speed devices to detect the speed of moving vehicles Figure In the laboratory scientists and engineers can use electronic devices such as motion sensors to measure speed Average Speed Figure A laser speed device can accurately measure the speed of an oncoming vehicle average speed vav the total distance travelled divided by the total time taken to travel that distance The average speed of a moving object is the total distance travelled divided by the total time elapsed You are probably familiar with the speedometer of a passenger vehicle which tells the speed of the vehicle in kilometres per hour km h However the SI unit for speed is metres per second m s You do not need a special device like a police speed device to measure speed If you know the distance travelled and the time it took an object to travel that distance you can calculate the average speed of the object using the equation vav Dd Dt where vav is the average speed d is the distance travelled and t is the change in time Like distance speed is a scalar quantity In the following Tutorial we will determine the average speed of an object using this equation Investigation Tutorial Calculating Average Speed The following Sample Problems will demonstrate how to use the equation for average speed Watch Your Speed p In this Investigation you will use the average speed equation to determine average speed in a study of vehicles passing an observation point Sample Problem Determining Average Speed Your dog runs in a straight line for a distance of m in s What is your dog s average speed Given d m t s Required vav Analysis vav Solution vav Dd Dt Dd Dt m s vav m s Statement Your dog s average speed is m s Sample Problem Determining the Distance Travelled by a Ball Moving at Constant Speed A baseball rolls along a flat parking lot in a straight line at a constant speed of m s How far will the baseball roll in s Given vav m s t s Required d Chapter Motion in a Straight Line NEL ---- SPLITTER -- Analysis vav Dd vavDt Solution Dd vavDt Dd m a Dd Dt m b s s LEARNiNg TIP Rounding in Calculations As a general rule round final answers to the same number of significant digits as the given value with the fewest significant digits Take extra care when rounding digits with multiple parts You will see in this book that extra digits are carried in intermediate calculations For more help with rounding refer to the Skills Handbook Statement The ball will roll m in s Practice A paper airplane flies m in s What is the airplane s average speed ans m s T I A cheetah can run at a maximum speed of km h or m s How far can a cheetah run in s T I ans m How long does it take a rock to fall through m of water if it falls at a constant speed of m s T I ans s research This Searching for Speeders Skills Researching Analyzing Identifying Alternatives Defending a Decision Communicating SKILLS HANDBOOK A A laser speed device is used by police officers to measure the speed of moving vehicles This device sends a pulse of infrared laser light at the speed of light m s toward a moving vehicle The laser pulse reflects off the vehicle and returns to a sensor on the speed device A computer chip in the speed device determines the time it took for the pulse to travel to and from the moving vehicle The speed device uses one half of this very short time and the speed of light to calculate the distance to the moving vehicle The speed device s computer uses multiple distance readings to determine how the vehicle s distance is changing with time and then calculates the vehicle s speed Modern speed devices send thousands of pulses of light each second providing a high level of accuracy Conduct research to investigate how common laser speed devices are in the region where you live Investigate how speed affects the number of automobile collisions and fatalities in Canada Investigate alternative methods the police could use to determine the speed of a vehicle A Does the use of laser speed devices have an impact on the number of automobile collisions and fatalities in Canada K u B Do you feel that the use of laser speed devices is the preferred way for police to monitor the speed of automobiles C C Laser speed devices and video recorders can now be used to capture the speed of a moving vehicle the vehicle s licence plate number the date and the time in the same image If these devices are set in a fixed position they can operate without the need for a police officer to be present Data can be collected electronically and speeders can be sent a ticket through the mail Do you support the use of devices like these in Ontario Justify your decision C A go To NELSoN SCiENCE NEL Speed and Velocity ---- SPLITTER -- Average velocity average velocity v av the total displacement or change in position divided by the total time for that displacement position time graph a graph describing the motion of an object with position on the vertical axis and time on the horizontal axis slope m a measure of the steepness of a line rise vertical change between two points on a line run horizontal change between two points on a line The average velocity of an object in motion is its total displacement or change in position divided by the total time taken for the motion Velocity describes change in position over time For instance a cyclist travelling east at a constant speed of m s has a velocity of m s E Since it has direction and magnitude average velocity is a vector quantity The SI unit for velocity is metres per second m s The symbol for average velocity is vav A position time graph is a graph that describes the motion of an object with position on the vertical axis and time on the horizontal axis Figure shows a position time graph for the motion of a rolling ball measured by students during an experiment Notice that the points on the graph form a straight line that moves upward from left to right Whenever an object is moving at a constant velocity the position time graph of that motion is a straight line You may recall from your mathematics studies that the slope of a line describes its steepness The symbol for slope is m Slope is determined as shown in Figure by comparing the magnitude of the rise the change between points on the y-axis and the magnitude of the run the change between the same points on the x-axis You can use this technique whether the graph passes through the origin or not for example in Figure the motion begins at a position of m E when t s For an object moving at a constant velocity so that its position time graph is a straight line the key relationship is this The slope of a position time graph gives the velocity of the object LEARNiNg TIP Rates of Change Average speed and average velocity are examples of rates of change an important concept in science that describes how quickly a quantity is changing Velocity is the rate of change of position which means that the more rapidly an object s position is changing the greater is the magnitude of its velocity The steeper the graph the greater is the object s displacement in a given time interval and the higher is its velocity This can be confirmed using the information in Figure Since the y-axis shows change in position Dd and the x-axis shows change in time t the formula for the slope of this graph can be rewritten as follows slope rise run d d m t t Dd or m Dt C -F -OP USB C -F -OP USB Position v Time for a Rolling Ball d m E rise m E t s Position v Time for a Rolling Ball d m E run s t s Figure Calculating the slope of a position time graph Chapter Motion in a Straight Line Figure A position time graph with non-zero initial position NEL ---- SPLITTER -- The average velocity of a moving object is given by the equation Dd v av Dt To determine the slope the average velocity from the zero point to the final data point for the x-axis and y-axis for the motion shown in Figure substitute the initial and final displacement and time values into the equation we just derived Dd v av Dt d d m t t m E m s s m m s E The velocity of the rolling ball is m s E Note that the slopes of the graphs shown in Figure and Figure are the same m s E The two motions are different in that the motion described by Figure started m away from the observer whereas the motion graphed in Figure had an initial position of m E from the observer Calculating average velocity from the slope of a position time graph is a very useful technique because average velocity is often difficult to measure directly However position can be easily measured with equipment such as tape measures motion sensors and laser speed devices Velocity a vector quantity is to speed a scalar quantity as displacement a vector quantity is to distance a scalar quantity The equation for average velocity should therefore look similar to the equation for average speed except that velocity and displacement are vectors Dd v av Dt where Dd is the change in position and t is the change in time during the given time interval This is the same equation as the one we just derived using the slope of a position time graph Note that t can also be described by the equation t t t Often we can simplify this equation by considering t the start time to be s In the following Tutorial we will use the average velocity equation to determine unknown values LEARNiNg TIP Calculations with Vectors In general you cannot divide one vector by another as dividing a direction has no meaning However if the directions of both vectors are the same you can disregard the direction and divide one magnitude by the other Tutorial Solving Problems Using the Equation for Average Velocity The equation for average velocity can be used to solve for any of the three variables in the average velocity equation when the other two are known In the following Sample Problems we will review solving equations for an unknown variable using the equation for average velocity Sample Problem Calculating the Average Velocity of an Object On a windy day the position of a balloon changes as it is blown m N away from a child in s What is the average velocity of the balloon We are given the change in time and the change in position of the balloon so we can solve for average velocity Given Dd m N Dt s Required vav Solution Statement The average velocity of the balloon is m s N Speed and Velocity Dd Analysis v av Dt Dd Solution v av Dt m N s v av m s N NEL ---- SPLITTER -- Sample Problem Calculating the Time for a Displacement to Occur A subway train travels at an average velocity of km h W How long will it take for the subway train to undergo a displacement of m W Given v av km h W Dd m W Required t Dd Analysis v av Dt a Since we are given the average velocity and the displacement of the subway train we can rearrange the average velocity equation to solve for the change in time Solution Before we solve this problem we must first make sure that all of the given values are converted to SI metric units This will require us to convert km h to metres per second We will do this by multiplying km h by a series of ratios equal to one We will use these ratios so that the units that we do not want kilometres and hours will cancel and we will be left with the units we do want metres and seconds Now that we have converted units we can use the average velocity equation to determine the change in time Dd v av Dt Dd Dt v av m W m s W W m b s vav m s W two extra digits carried min m km h W b a vav a ba b ba h min s km Dt s Statement It takes the subway train s to be displaced m W from its starting point Practice What is the average velocity of a soccer ball that is kicked and travels m E in s T I ans m s E How long will it take a cat to run m N at an average velocity of m s N T I ans s Motion with Uniform and Non-uniform velocity motion with uniform or constant velocity motion of an object at a constant speed in a straight line motion with non-uniform velocity accelerated motion motion in which the object s speed changes or the object does not travel in a straight line line It is the simplest type of motion that an object can undergo except for being at rest Note that both requirements constant speed and straight line must be met for an object s velocity to be uniform In contrast motion with non-uniform velocity is motion that is not at a constant speed or not in a straight line Motion with non-uniform velocity may also be called accelerated motion Table shows some examples of motion with uniform velocity You will learn more about accelerated motion in Section Motion with uniform or constant velocity is motion at a constant speed in a straight Table Examples of Uniform and Non-uniform Velocity Example A car travels down a straight highway at a steady km h A passenger on an amusement park ride travels in a circle at a constant speed of m s A parachutist jumps out of an aircraft after parachute opens Uniform velocity Non-uniform velocity Explanation The car is travelling at a constant speed in a straight line The passenger is travelling at a constant speed but not in a straight line She is travelling in a circle Before he opens the parachute the speed of the parachutist will increase due to gravity Once the parachute is opened his speed will become constant due to air resistance He will then fall at a constant speed in the same direction downwards before parachute opens Chapter Motion in a Straight Line NEL ---- SPLITTER -- determining Types of Motion from Position Time Graphs Recall that the slope of a position time graph gives the velocity of an object A position time graph that describes constant velocity must be a straight line This is because motion with constant velocity is motion at a constant speed Therefore the slope of the position time graph must also be constant which makes it a straight line Table shows five position time graphs that represent commonly occurring types of motion You will see these types of motion frequently in investigations By the end of this unit you should be able to identify the type of motion from the characteristics of its position time graph Table Interpreting Position Time Graphs Position time graph Graph A C -F - P USB Type of motion graph is a horizontal straight line the slope of a horizontal straight line is zero the object has a velocity of zero the object is at rest t he object is at a constant positive position relative to the reference position he object is stationary at a location to the t east of the reference position Example C -P - P USB d m E PHOTO TO COME t s Graph B C -F - P USB d m E t s graph is a horizontal straight line the slope of a horizontal straight line is zero the object has a velocity of zero the object is at rest t he object is at a constant negative position relative to the reference position he object is stationary at a location to the t west of the reference position C -P - P USB PHOTO TO COME Graph C C -F - P USB graph is a straight line with positive slope traight lines with non-zero slopes always s represent constant non-zero velocity rom the y-axis we know the object is f moving eastward he object s velocity can be determined from t the slope of the graph rise divided by run C -P - P USB d m E t s Graph D C -F - P USB t s raph is a straight line with positive slope g which represents constant positive velocity rom the y-axis we know the object is f moving eastward he object s velocity can be determined from t the slope of the graph ince graph D has the steeper slope we s can conclude that this object has a greater velocity than the object described by graph C raph is a straight line which represents g constant velocity the slope of the graph is negative he object s velocity can be determined from t the slope of the graph ote that the direction for position on the n y-axis is given by a vector with direction E he negative slope indicates that the object is t moving westward C -P - P USB d m E Graph E C -F - P USB C -P - P USB USB roup owle d m E t s NEL Speed and Velocity ---- SPLITTER -- Mini investigation Bodies in Motion Skills Controlling Variables Performing Observing Analyzing SKILLS HANDBOOK A Motion sensors are devices that send out ultrasonic wave pulses When some of these waves re ect off an object the waves return to the motion sensor A computer can then analyze the data from the returning waves and generate real-time position time graphs Using motion sensors can help you understand position time graphs Equipment and Materials motion sensor computer or computer interface graph paper pencil Connect the motion sensor to a computer or computer interface Place the motion sensor on a lab bench Make sure that you have about m of free space in front of the motion sensor You will be using your body to generate position time data with the motion sensor Before you begin sketch position time graphs to show the overall shape and features of the graphs that you predict the motion sensor will generate for each of the following scenarios slow constant speed speeding up fast constant speed slowing down Using the motion detector generate real-time position time graphs for each scenario in Step using body motions A Compare your predictions with the results Explain any differences T i C B If there is time use the motion sensor to generate graphs that resemble as many letters of the alphabet as you can T i C Summary Average speed is equal to the total distance travelled divided by the time taken for the motion A position time graph describes motion graphically The slope of the position time graph gives the velocity of an object Average velocity is equal to the total displacement divided by the time taken for the motion In other words velocity describes change in position over time Motion with uniform or constant velocity is motion at a constant speed in a straight line Objects that are undergoing constant velocity have a position time graph that is a straight line Questions When you are solving a problem how do you know if you are given a speed value or a velocity value K U De ne motion with uniform velocity in your own words K U C Give two real-life examples each of motion with uniform velocity and motion with non-uniform velocity A Determine the velocity for the motion described by the graph in Figure T i Copy and complete Table in your notebook Table T i v av Dd m S Dt s m s E m s N m E s What is the displacement of a horse that runs at a velocity of m s S for s T i How many seconds would it take a car travelling at km h to travel a distance of m T i What is the velocity in metres per second of a Canadian Forces CF- ghter jet that travels km S in min T i t s C -P - P USB NEL d m W Figure Chapter Motion in a Straight Line ---- SPLITTER -- Acceleration Some theme parks have rides in which you are slowly carried up in a seat to the top of a tower and then suddenly released On the way down your arms and hair may fly upward as the velocity of your seat increases The thrill of this sudden change in motion can frighten and exhilarate you all at once An even bigger thrill ride however is to be a pilot in a jet being launched from the deck of an aircraft carrier These giant ships use catapults to move kg jets from rest km h to km h in just s While it is true that objects sometimes move at constant velocity in everyday life usually the velocities we observe are not constant Objects that experience a change in velocity are said to be undergoing acceleration Acceleration describes how quickly an object s velocity changes over time or the rate of change of velocity We can study acceleration using a velocity time graph which similar to the position time graph has time on the horizontal axis but velocity rather than position on the vertical axis Velocity time graphs are particularly useful when studying objects moving with uniform velocity zero acceleration or uniform acceleration velocity changing but at a constant rate The velocity time graphs for both uniform velocity and uniform acceleration are always straight lines By contrast the position time graph of an accelerated motion is curved Table shows the position time graphs of objects moving with accelerated motion acceleration a av how quickly an object s velocity changes over time rate of change of velocity Figure Sudden changes in velocity are part of the thrill of midway rides C -P -OP USB velocity time graph a graph describing the motion of an object with velocity on the vertical axis and time on the horizontal axis Table Interpreting Position Time Graphs Position time graph Graph A C -F - P USB Type of motion graph is a curve n any graph that curves the slope or steepness of the o graph changes from one point on the graph to another ince the slope in graph A is constantly changing the s velocity is not constant ince the graph lies above the x-axis and its slope is s increasing the velocity of the object is also increasing speeding up in a positive or eastward direction Example C -P - P USB d m E t s Graph B t s C -F - P USB graph is a curve n any graph that curves the slope or steepness of the o graph changes from one point on the graph to another ince the slope in graph B is constantly changing the s velocity is not constant ince the graph lies below the x-axis and its slope is s negative but getting steeper the object is speeding up as it moves in a negative or westward direction P USB Group d m E NEL Acceleration ---- SPLITTER -- Table continued Position time graph Graph C C -F - P USB Type of motion graph is a curve n any graph that curves the slope changes from o one point on the curve to another ince the slope in graph C is constantly changing the s velocity is not constant ince the graph lies above the x-axis and its slope is s decreasing the velocity of the object is also decreasing slowing down in a positive or eastward direction Example d m E C -P - P USB t s Graph D C -F - P USB t s graph is a curve n any graph that curves the slope or steepness of the o graph changes from one point on the graph to another ince the slope in graph D is constantly changing the s velocity is not constant ince the graph lies below the x-axis and its slope is s negative and getting shallower the object is slowing down as it moves in a negative or westward direction d m E Determining Acceleration from a Velocity Time Graph Figure shows a velocity time graph for a skateboard rolling down a ramp Notice that the line of the graph goes upward to the right and has x-intercept and y-intercept of zero We can calculate the slope of the graph in Figure using the equation slope rise run Dv m s m Dt s SB p e Since acceleration describes change in velocity over time this suggests that the average acceleration of the skateboard is given by the equation Dv a av Dt C -F - P USB SB p e LEarNiNg Tip Slope and Area of Velocity Time Graphs The slope of a velocity time graph gives the acceleration of the object The area under a velocity time graph gives the displacement of the object Why is displacement related to the area under a velocity time graph One way to think about it is this the greater the velocity during a given time interval the greater the area under the graph and the greater the displacement over that time interval v m s S Velocity v Time for a Skateboard Rolling Down a Ramp t s Figure Velocity time graph for a skateboard rolling down a ramp NEL Chapter Motion in a Straight Line ---- SPLITTER -- In other words The slope of a velocity time graph gives the average acceleration of an object Acceleration over a time interval that is average acceleration is given by the equation average acceleration change in velocity change in time Dv aav Dt vf vi or aav Dt LEARNiNg TIP Square Seconds What is a square second Good question When we write acceleration units as m s we are not implying that we have measured a square second This is simply a shorter way of expressing the derived unit You can also read the unit as metres per second per second describing how many metres per second of velocity are gained or lost during each second of acceleration Recall that the SI unit for velocity is metres per second m s and the SI unit for time is seconds s The SI unit for acceleration is a derived unit A derived SI unit is a unit created by combining SI base units We can derive the units for acceleration by dividing a velocity unit m s by a time unit s as follows units of acceleration units of velocity per second m s s m s s m s vf vi for When you are given values for any three variables in the equation a av Dt acceleration you can solve for the missing variable Tutorial Calculating Acceleration Sample Problem Analysis To calculate the average acceleration which is the slope of the graph we use the defining equation in the form that includes v i v f t i and t f Dv aav Dt vf vi tf ti vf vi Solution aav tf ti aav m s S m s S m s s s What is the acceleration of the skateboard in Figure Consider the motion between s and s Given v i m s vf m s S t i s t f s Required aav Statement The skateboard is accelerating down the ramp at m s S What would the acceleration be if the same velocity change of m s S took place over a time interval of only s aav m s S Dv aav Dt m s S s The time interval is shorter so a more rapid acceleration occurs NEL Acceleration ---- SPLITTER -- Sample Problem A bullet is found lodged deeply in a brick wall As part of the investigation a forensic scientist is experimenting with a rifle that was found nearby She needs to determine the acceleration that a bullet from the rifle can achieve as a first step in linking the rifle to the bullet During a test firing she finds that the rifle bullet accelerates from rest to m s E in s as it travels down the rifle s barrel What is the bullet s average acceleration Given v i m s v f m s E Dt s Required a av vf vi Analysis a av Dt vf vi Solution a av Dt Statement The acceleration of the bullet is m s E dilemma we can use a technique from Section We will change a vector subtraction problem into a vector addition problem Recall that negative N is the same as positive S m s S m s S a av s aav m s S m s S s a av m s E m m E s s s Sample Problem When a hockey player hits a hockey puck with his stick the velocity of the puck changes from m s N to m s S over a time interval of s What is the acceleration of the puck Given v i m s N v f m s S Dt s At this point it would appear that we have a dilemma In the numerator of the fraction we must subtract a vector with a direction N from a vector with a direction S To solve this Required a av vf vi Analysis a av Dt vf vi Solution a av Dt m s S m s N a av s Statement The hockey puck s acceleration is m s S Notice that the initial velocity of the puck is north while its final velocity is south The acceleration is in the opposite direction to the initial motion so the puck slows down and comes to rest It then continues accelerating south increasing its velocity to m s S This is why the final velocity is due south Sample Problem can also be solved by using a vector scale diagram Practice A catapult accelerates a rock from rest to a velocity of m s S over a time interval of s What is the rock s average acceleration T I ans m s S As a car approaches a highway on-ramp it increases its velocity from m s N to m s N over s What is the car s average acceleration T I m s N A squash ball with an initial velocity of m s W is hit by a squash racket changing its velocity to m s E in s What is the squash ball s average acceleration T I m s E You can use both the defining equation for average acceleration and velocity time graphs to determine other information about the motion of an object besides acceleration itself In Tutorial you will use the equation in a different form to determine a different quantity Tutorial introduces the idea of determining displacement via the area under a velocity time graph Chapter Motion in a Straight Line NEL ---- SPLITTER -- Tutorial Solving the Acceleration Equation for Other Variables In the following Sample Problem we will explore how to solve the defining acceleration equation for other variables Sample Problem Solving the Acceleration Equation for Final Velocity A racehorse takes s to accelerate from a trot to a gallop If the horse s initial velocity is m s W and it experiences an acceleration of m s W what is the racehorse s velocity when it gallops Given Dt s v i m s W a av m s W Required v f vf vi Analysis a av Dt Solution Rearrange the equation for acceleration to solve for the final velocity a av Dt v f v i v f a av Dt v i Statement When it gallops the racehorse has a velocity of m s W m s W m s W vf m s W a m m W b s W s s Practice How long does it take a radio-controlled car to accelerate from m s W to m s W if it experiences an average acceleration of m s W T I ans s A speedboat experiences an average acceleration of m s W If the boat accelerates for s and has a final velocity of m s W what was the initial velocity of the speedboat T I ans m s W Tutorial Determining Displacement from a Velocity Time Graph By further analyzing the velocity time graph shown in Figure on page we can determine even more information about the motion of the skateboard Figure has the same data plotted as Figure except that the area under the line is shaded The area under a velocity time graph gives the displacement of the object Note how the area under the straight line in Figure forms a triangle The area A of a triangle is determined by the equation A bh Dd bh where b is the length of the base of the triangle and h is the height of the triangle We can determine the displacement of the skateboard from Figure by calculating the area of this triangle v m s S base b s t s height h m s S Velocity v Time for a Skateboard Rolling Down a Ramp C -F - P USB Figure Velocity time graph showing the area underneath the line NEL Acceleration ---- SPLITTER -- Sample Problem Determining Displacement from a Velocity Time Graph What is the displacement represented by the graph in Figure on the previous page Given b s h m s S Required Dd Analysis Dd bh Statement The object was displaced m S in s m Solution Dd s a S b s Dd m S Sample Problem Determining Displacement from a More Complex Velocity Time Graph What is the displacement represented by the graph in Figure over the time interval from s to s C -F - P USB v m s S b Velocity v Time for a Complex Motion h s t s w m s S l s Figure Velocity time graph showing more complex motion The graph in Figure is a more complex velocity time graph than we worked with in Sample Problem However the displacement of the object can still be determined by calculating the area under the velocity time graph To calculate this displacement we will need to break the area under the line into a rectangle and a triangle and add these two areas together We know the area A of a triangle is given by Atriangle bh The area of a rectangle is its length l multiplied by its width w or Arectangle lw hysics U Given b s h m s S l s w m s S Required Dd Analysis Dd Atriangle Arectangle Solution Dd Atriangle Arectangle bh lw d oved Statement The object has travelled m S after s C -F -OP USB CrowleArt Group Practice Deborah Crowle Determine the displacement represented by the graph in Figure over the following rd pass intervals T I time a from s to s ans m S b from s to s ans m S m S m S Dd m S m m s a S b s a S b s s Chapter Motion in a Straight Line NEL ---- SPLITTER -- Mini Investigation Motion Simulations Skills Predicting Performing Observing Analyzing Communicating SKILLS HANDBOOK A In this investigation you will use a computer simulation for four different motion scenarios and then analyze the scenarios by graphing Equipment and Materials computer access graphing paper Go to the Nelson website and find the link for this Mini Investigation Go to the simulation and take a few minutes to familiarize yourself with how it operates You will be asked to run this simulation for each of the following scenarios a positive velocity value a negative velocity value a negative initial position and a positive velocity value a negative initial position and a negative velocity value A Before you run these scenarios write a brief statement and draw a sketch to predict how each graph will appear T I C B After you have run each scenario sketch the resulting position time and velocity time graphs from the actual data you collected T I C C Compare your sketches from Question A to the graphs that were generated when you ran each scenario Explain any discrepancies or misconceptions that you may have had T I go To NELSoN SCiENCE C -P - P USB Figure Launch of NASA s Mars Pathfinder mission Instantaneous velocity and Average velocity The velocity of any object that is accelerating is changing over time In motion with uniform acceleration the velocity of an object changes at a constant uniform rate During a launch Figure a spacecraft accelerates upward at a rapid rate NASA personnel may need to determine the velocity of the spacecraft at specific points in time They can do this by plotting position and time data on a graph and determining the spacecraft s instantaneous velocity Instantaneous velocity or v inst is the velocity of an object at a specific instant in time By comparison average velocity v av is determined over a time interval Both types of velocity are rates of change of position but they tell us different things about the motion of an object motion with uniform acceleration motion in which velocity changes at a constant rate instantaneous velocity vinst the velocity of an object at a specific instant in time Tutorial Determining Instantaneous and Average Velocity Figure on the next page shows a position time graph of an object that is undergoing uniform acceleration Moving along the curve the slope of the curve progressively increases From this we know that the speed of the object is constantly increasing To determine the instantaneous velocity of the object at a specific time we must calculate the slope of the tangent of the line on the position time graph at that time A tangent is a straight line that contacts a curve at a single point and extends in the same direction as the slope of the curve at the point A plane mirror can be used to draw a tangent to a curved line Place the mirror as perpendicular as possible to the line at the point desired Adjust the angle of the mirror so that the real curve merges smoothly with its image in the mirror which will occur when the mirror is perpendicular to the curved line at that point Draw a line perpendicular to the mirror to obtain the tangent to the curve NEL Acceleration ---- SPLITTER -- C -F - P USB d m E Position v Time for Motion with Uniform Acceleration rise run t s Figure Position time graph with non-constant velocity We can use Figure to determine the instantaneous velocity of the object at any specific point in time To determine the instantaneous velocity we must determine the slope of the tangent of the line on the position time graph at that specific time Sample Problem Determining Instantaneous Velocity Consider the point on the curve in Figure at s on the x-axis What is the instantaneous velocity of the object at this time Given t s position time graph Required v inst Solution m Physics Analysis v inst is equal to the slope m of the tangent to the U Dd curve at t s so m Dt C -F -OP USB In Figure Group CrowleArt the tangent to the point on the curve at t s has been extended until it crosses a convenient grid line Deborah Crowle Calculate the slope of the tangent to determine the nd pass instantaneous velocity d roved Sample Problem Determining Average Velocity from What is the average velocity of the object in Figure over the time interval from s to s Given d m d m E t s t s Required v av Statement The instantaneous velocity of the object at s is m s E Since the object is accelerating if we had calculated the slope of the tangent at time t s the velocity would have been smaller in magnitude and if we had calculated the slope of the tangent at time t s the velocity would have been greater We can also use Figure to determine the average velocity of the object v inst m s E m E s a Position Time Graph d d Solution v av t t Analysis Recall that average velocity is the total displacement of an object divided by the total time taken Dd v av Dt d d v av t t vav m s E m E m s s Statement The average velocity of the object over the time interval from s to s is m s E Chapter Motion in a Straight Line NEL ---- SPLITTER -- Practice a Determine the instantaneous velocity at t s for the graph shown in Figure ans m s E b Determine the instantaneous velocity at t s for the graph shown in Figure ans m s E c Compare your answers to a and b with the solution to Sample Problem above Is it possible that the object is moving with constant acceleration Explain T I a Determine the instantaneous velocity at t s for the graph shown in Figure ans m s E b Determine the average velocity from t s to t s for the graph ans m s E c Notice that for the motion described in Figure t s is the midpoint in time Write a statement describing the relationship that exists between the average velocity and the instantaneous velocity at the midpoint in time when an object is accelerating uniformly T I C -F - P USB C d m E Figure Position v Time for Accelerated Motion t s Note that in Tutorial the average velocity over the first s of the motion was less than the instantaneous velocity at s For motion with non-uniform velocity but uniform acceleration average and instantaneous velocities are not necessarily equal The only situation where the average velocity and the instantaneous velocity are the same is at the midpoint in the time interval Summary Acceleration describes change in velocity over time The slope of a velocity time graph gives the acceleration of the object whose motion it describes The area under a velocity time graph gives the displacement of the object whose motion it describes The instantaneous velocity of an object is its velocity at a specific instant in ario Physics U time It is equal to the slope of the tangent to the position time graph at that instant in time For motion with non-uniform velocity average and instantaneous velocities C -F -OP USB are not necessarily equal CrowleArt Group Deborah Crowle NEL rd pass s Acceleration ---- SPLITTER -- Questions a Describe the motion of the object in all three segments of the graph shown in Figure b Calculate the average acceleration of the object in all three segments of the graph in Figure c Calculate the total displacement of the object from s to s from s to s and from s to s T I C C -F - P USB Describe three characteristics that an accelerating object may exhibit Give a real-world example of each characteristic K u C Describe in your own words how you would determine the acceleration of an object from a velocity time graph K u C Describe in your own words how you would determine the displacement of an object from a velocity time graph K u C Determine the average acceleration described by each of the following graphs T I C -F a- P USB Velocity v Time for Accelerated Motion a v m s E v m s W Figure Velocity v Time for Complex Motion t s t s b v m s E Velocity v Time for Accelerated Motion C -F b- P USB What is the average acceleration of a sports car that increases its velocity from m s W to m s W in s T I If a child on a bicycle can accelerate at an average rate of m s how long would it take to increase the bicycle s velocity from m s N to m s N T I a While approaching a red light a student driver begins to apply the brakes If the car s brakes can cause an average acceleration of m s S and it takes s for the car to come to rest what was the car s initial velocity b What is the significance of the direction of the initial velocity and that of the acceleration T I What is the average acceleration of a tennis ball that has an initial velocity of m s E and a final velocity of m s W if it is in contact with a tennis racket for s T I a Determine the instantaneous velocity at t s in Figure b Determine the average velocity of the motion depicted in Figure T I C -F - P USB d m W Figure t s Position v Time for Accelerated Motion t s c v m s E Velocity v Time for Accelerated Motion U Ontario Physics U C -F -OP USB FN CrowleArt Group CO Deborah Crowle nd pass Pass Approved Not Approved C -F c- P USB rowleArt Group t s Deborah Crowle One of your classmates makes the following statement If th pass -F a-OP USB an object has an initial velocity of m s N and a final velocity of m s S this object has clearly not accelerated as it is travelling at a constant speed Write an email to this student explaining why this statement is incorrect K u C U -F b-OP USB owleArt Group eborah Crowle h pass Chapter Motion in a Straight Line NEL ---- SPLITTER -- Comparing Graphs of Linear Motion Cheetahs are adapted for speed they are the fastest land animals They can accelerate at faster rates than most sports cars Figure Cheetahs have been measured accelerating at rates greater than m s To put this in perspective a sports car can accelerate at approximately m s In fact cheetahs are capable of accelerating from rest to m s in only three strides You have already seen how position time and velocity time graphs can be used to analyze the linear motion of objects In this section we will introduce acceleration time graphs and use all three types of graphs to analyze motion in more detail C -P -OP USB Figure Cheetahs have the greatest acceleration of any animal Acceleration Time Graphs Earlier in this chapter you learned how to find the displacement or change in position of an object by determining the area under a velocity time graph In a similar way we can determine the change in velocity of an object from the area under an acceleration time graph which has acceleration on the vertical axis and time on the horizontal axis Consider the acceleration time graph in Figure which shows the motion of a cheetah The points plotted on this graph lie along a horizontal straight line with a non-zero y-intercept The acceleration is a constant m s so this graph represents uniform acceleration C -F -OP USB acceleration time graph a graph describing motion of an object with acceleration on the vertical axis and time on the horizontal axis a m s W Acceleration v Time for Motion with Uniform Acceleration t s Figure Acceleration time graph showing motion with uniform acceleration NEL Comparing Graphs of Linear Motion ---- SPLITTER -- Investigation Uniform Velocity p In this investigation you will use a motion sensor to generate different types of motion graphs for an object moving with uniform velocity and analyze these graphs If we calculate the area under the acceleration time graph in Figure from s to s we will be determining the change in velocity of the object from t s to t s A lw A m s W s a m W b s Since the units are metres per second the area we calculated represents a change in velocity The area under an acceleration time graph represents the change in velocity of an object If the initial velocity of the cheetah is zero the object is at rest the final velocity is equal to the change in velocity m s W If the initial velocity is m s W however then the graph tells us that the final velocity is m s W m s W m s W The graph does not tell us what the initial and final velocities are it just tells us the change in velocity that occurs in the time interval Relationships among Linear Motion Graphs Investigation Motion Down a Ramp p In this investigation you will use a motion sensor and different types of motion graphs to analyze the motion of an object rolling down a ramp Graphical analysis is one of the most powerful analytical tools available to physicists In studying the motion of objects analyzing position time velocity time and acceleration time graphs can help us gain insight into real-life events such as the motion of the cheetah shown in Figure This is particularly important because most objects in nature do not come equipped with a speedometer Figure compares the three types of graphs of linear motion All three graphs represent the same type of motion uniform acceleration Nevertheless the three graphs look very different When analyzing a motion graph you may read information directly from the graph or determine further information by calculating the slope or the area of the graph C -F - P USB slope slope t s t s a m s v m s d m area area t s Figure Position time velocity time and acceleration time graphs of the same motion Chapter Motion in a Straight Line NEL ---- SPLITTER -- Tutorial Creating One Type of Motion Graph from Another By using the information in Figure we can analyze an acceleration time graph further and get more information about the motion it describes Sample Problem Creating a Velocity Time Graph from an Acceleration Time Graph Use the acceleration time graph in Figure to generate velocity and time data for the object Then use these data to plot a velocity time graph Step To generate the velocity time data first calculate the area under the graph for several time points in Figure Since the line is horizontal we use the formula for the area of a rectangle A lw For Figure l t s w a m s W and A v m s W so in calculating A lw we are actually calculating v Da Dt C -F -OP USB Step Plot the data to create a velocity time graph Figure C -F -OP USB v m s W Velocity v Time for Motion with Uniform Acceleration a m s W Acceleration v Time for Motion with Uniform Acceleration t s Figure Velocity time solution graph Figure shows the resulting graph It is an increasing straight line with a zero intercept It describes precisely the same motion that was described by the acceleration time graph in Figure Both graphs describe uniform acceleration t s Figure Using an acceleration time graph to create other motion graphs Step Table shows the calculations for the area under the curve at s intervals from t Physics U Ontario s to t s Acceleration a m s W Pass m v a W b s s Approved Table Calculating the Velocity at Various Time Points in Figure Time t s U C -F -OP USB CrowleArt Group Deborah Crowle rd pass m v a W b s s m v a W b s s m v a W b s s m v a W b s s Not Approved m v a W b s s FN Equation CO v Da Dt C -F -OP USB Velocity CrowleArt Group v m s W Deborah Crowle rd pass NEL Comparing Graphs of Linear Motion ---- SPLITTER -- Sample Problem Creating an Acceleration Time Graph from a Velocity Time Graph Use the velocity time graph shown in Figure to plot the corresponding acceleration time graph C -F -OP USB slope v m s W Velocity v Time for Motion with Uniform Acceleration rise run Dv a av Dt a av m s W Step Use this value to create an acceleration time graph Figure Acceleration v Time for Motion with Uniform Acceleration C -F -OP USB m s W s a m s W t s Figure Acceleration time solution graph t s Figure shows the corresponding acceleration time graph This graph shows a horizontal straight line with a y-intercept of m s W Note If a velocity time graph is not a straight line you will need to determine the slope of the tangent for each time data point and then use these data to plot the corresponding acceleration time graph Figure Given velocity time graph Step The data plotted on the velocity time graph in Figure form an increasing straight-line graph with a zero intercept You can determine acceleration from a velocity time graph by calculating its slope Since the velocity time graph in Figure is a straight line its slope does not change So Physicscalculate the Ontario we can U slope or acceleration over any time interval C -F -OP USB FN C -F -OP USB CrowleArt Group CO Generate position time and acceleration time data Deborah Crowle representing the motion of the object shown in Figure nd pass Pass Use the data to plot the corresponding position time and Approved acceleration time graphs T I C -OP USB Not Approved leArt Group rah Crowle pass Practice Velocity v Time for Motion with Uniform Acceleration v m s N t s Figure Chapter Motion in a Straight Line NEL ---- SPLITTER -- Summary The area under an acceleration time graph gives the velocity of the object Given one type of motion graph you can read or calculate data from it in order to construct a different type of graph Questions Copy and complete Table in your notebook by adding a check mark in each column that applies K U Table How do you determine position velocity velocity velocity acceleration acceleration Read information from graph Given a position time graph position time graph velocity time graph acceleration time graph velocity time graph acceleration time graph Take the slope Find the area C -F -OP USB From the velocity time graph in Figure generate position time data and then plot the corresponding position time graph assuming the initial position is m T I C C -F -OP USB d m S Position v Time for Accelerated Motion v m s S Velocity v Time for Complex Motion Figure t s t s Figure Use the data in the velocity time graph shown in Figure to plot the corresponding acceleration time graph T I C C -F -OP USB v m s S Consider the position time graph shown in Figure T I a What is the position of the object at t s b What is the instantaneous velocity of the object at t s c What is the average velocity for the object s motion from s to s Figure Velocity v Time for Accelerated Motion t s NEL U Ontario Physics U C -F -OP USB FN CrowleArt Group CO Comparing Graphs of Linear Motion ---- SPLITTER -- Five Key Equations for Motion with Uniform Acceleration Graphical analysis is an important tool for physicists to use to solve problems Sometimes however we have enough information to allow us to solve problems algebraically Algebraic methods tend to be quicker and more convenient than graphical analysis For example if you want to determine how far a passing vehicle would travel in a given amount of time you could perform an experiment using a motion sensor You would collect position time data with the motion sensor and then plot the data on a graph From the graph you could then measure how far the vehicle has gone in a given amount of time However if you were in the vehicle you would simply use the vehicle s speedometer to determine the speed of the vehicle Knowing the speed of your vehicle you could easily determine how far it would travel in a given time interval Dd using the equation vav As you can see the best way to solve a problem is usuDt ally determined by the information that is available to you To be able to solve problems related to motion with uniform acceleration in which the velocity may change but the acceleration is constant we need to derive algebraic equations that describe this type of motion We will start with equations that we have already used in previous sections C -F -OP USB vf v m s E area vi area t s Figure A velocity time graph for an object undergoing uniform acceleration A Displacement Equation for Uniformly Accelerated Motion The velocity time graph in Figure shows a straight line with a non-zero intercept This graph is a non-horizontal straight line showing that the object is undergoing uniform or constant acceleration In other words the velocity is increasing at a uniform or constant rate We know that to determine the displacement of this object from the velocity time graph we must determine the area under the line For the graph in Figure we must determine the area of a rectangle and a triangle Dd A triangle A rectangle bh lw Dt v f v i Dtv i v fDt v iDt v iDt v fDt v iDt vf vi Dd a bDt Equation LEArNING TIp Interpreting Areas Under a Motion Graph Notice that the rectangular area green in Figure represents the displacement the object would have undergone had it continued at constant velocity v i The triangular area blue represents the extra displacement the object experienced due to its acceleration We can use Equation to determine the displacement of an object that is undergoing uniform acceleration Equation is very similar to an equation that we previ ously developed from the defining equation for average velocity Dd v avDt We can relate the average velocity to the initial and final velocities by the equation vf vi v av a b vf vi Then substituting v av in place of a b in Equation gives Dd v avDt directly NEL Chapter Motion in a Straight Line ---- SPLITTER -- As you will see Equation can help us to solve many motion problems However in some situations we will not know the initial velocity the final velocity and the time interval for a given scenario We could use the defining equation for acceleration in a two-step process but this tends to be difficult To simplify things we can derive a number of other motion equations that will allow us to solve problems in one step Additional Motion Equations vf v Consider the defining equation for acceleration a av Dt v f v i a avDt Equation If we rearrange this equation to solve for final velocity v f we get Equation You may use Equation in problems that do not directly involve displacement If we substitute the expression vi a avDt from Equation into Equation we get v f v i a avDt Equation vf vi Dd a bDt Equation v i a avDt v i Dt v i a avDt Dt Dd v i Dt a av Dt Equation This is Equation which allows you to determine the displacement of an object moving with uniform acceleration given a value for acceleration rather than a final velocity The Five Key Equations of Accelerated Motion Table shows the five key equations of accelerated motion You should be able to solve any kinematics question by correctly choosing one of these five equations You have seen how the first three are developed We will leave the others to be developed as an exercise Table The Five Key Equations of Accelerated Motion Equation vf vi Dd a bDt v f v i a avDt Variables found in equation Dd Dt v f v i a av Dt v f v i Dd a av Dt v i Dd aav vf vi Dd a av Dt v f Variables not in equation a av Dd vf Dt vi Equation Equation Equation Equation Equation Dd v iDt a avDt vf vi aavDd Dd v fDt a avDt NEL Five Key Equations for Motion with Uniform Acceleration ---- SPLITTER -- Tutorial Using the Five Key Equations of Accelerated Motion The following Sample Problems will demonstrate how to choose equations and solve problems involving the five key motion equations Sample Problem A sports car approaches a highway on-ramp at a velocity of m s E If the car accelerates at a rate of m s E for s what is the displacement of the car C -F -OP USB ai Given v i m s E a av m s E Dt s Required Dd Analysis Our first task is to determine which of the five equations of accelerated motion to use Usually you can solve a problem using only one of the five equations We simply identify which equation contains all the variables for which we have given values and the unknown variable that we are asked to calculate In Table we see that Equation has all the given variables and will allow us to solve for the unknown variable Dd v iDt a avDt Solution Dd v iDt a avDt Statement During the s time interval the car is displaced m E a m m b a b s s m a b s m s m s m E m E Dd m E a m m E b s a E b s s s Sample Problem A sailboat accelerates uniformly from m s N to m s N at a rate of m s N What distance does the boat travel Given vi m s vf m s aav m s Required d Analysis In Table we see that Equation will allow us to solve for the unknown variable First we rearrange the equation to solve for d vf vi aavDd vf vi aavDd vf vi Dd aav Solution Dd vf vi aav d m Statement The boat travels a distance of m Sample Problem A dart is thrown at a target that is supported by a wooden backstop It strikes the backstop with an initial velocity of m s E The dart comes to rest in s a What is the acceleration of the dart Given v i m s E v f m s Dt s Required a av b How far does the dart penetrate into the backstop vf vi Solution a av Dt m m E s s s Solution For a Analysis We may use the defining equation for acceleration vf vi a av Dt Notice that the acceleration is in the opposite direction to the initial motion This must be true in order for the velocity of the dart to decrease to zero as it comes to rest If the acceleration were in the same direction as the initial velocity the final velocity would be greater than the initial velocity Statement The acceleration of the dart is m s W m s E a av m s W Chapter Motion in a Straight Line NEL ---- SPLITTER -- Practice To solve b we have sufficient information to solve the problem using any equation with displacement in it Generally speaking in a two-part problem like this it is a good idea to try to find an equation that uses only given information Then if we have made an error in calculating the first part acceleration our next calculation would be unaffected by the error Therefore we will use Equation to solve b since it can be solved using only given information Given v i m s E v f m s Dt s Required Dd vf vi Analysis Dd a bDt vf vi Solution Dd a bDt m m E s s s Statement The displacement of the dart into the backstop is m E Dd m E A football player initially at rest accelerates uniformly as she runs down the field travelling m E in s What is her final velocity T I ans m s E A child on a toboggan sits at rest on the top of a tobogganing hill If the child travels m downhill in s while accelerating uniformly what acceleration does the child experience T I ans m s downhill Summary The five key equations of accelerated motion listed in Table on page apply to motion with uniform constant acceleration They involve the variables for displacement initial velocity final velocity acceleration and time interval When solving uniform acceleration problems choose which equation s to use based on the given and required variables of the problem Questions A car accelerates from rest at a rate of m s N What is the displacement of the car at t s T I An astronaut is piloting her spacecraft toward the International Space Station To stop the spacecraft she fires the retro-rockets which cause the spacecraft to slow down from m s E to m s in s T I a What is the acceleration of the spacecraft b What is the displacement of the spacecraft when it comes to rest A helicopter travelling at a velocity of m s W accelerates uniformly at a rate of m s E for s What is the helicopter s final velocity T I Two go-carts A and B race each other around a km track Go-cart A travels at a constant speed of m s Go-cart B accelerates uniformly from rest at a rate of m s Which go-cart wins the race and by how much time T I A boat increases its speed from m s to m s over a distance of m What is the boat s acceleration T I Within s of liftoff a spacecraft that is uniformly accelerating straight upward from rest reaches an altitude of m up T I a What is the spacecraft s acceleration b At what velocity is the spacecraft travelling when it reaches this altitude Derive Equation and Equation in Table on page by substituting other expressions T I NEL Five Key Equations for Motion with Uniform Acceleration ---- SPLITTER -- Acceleration Near Earth s Surface During a basketball game the player takes the ball and shoots it toward the basket Figure The ball briefly skims the rim then drops through the net to the floor The player makes the basket because of her skill with a little bit of help from the force of gravity Gravity causes all objects to accelerate toward Earth s centre If you have accidentally dropped an object such as a glass you have directly experienced how significant the effect of gravity is C -P -OP USB LEArNING TIp Describing Vertical Motion Directions Vertical motion problems are vector questions that require directions to be indicated Directions can be simplified by defining up as positive and down as negative This means that the acceleration due to gravity should be negative in your calculations Figure Earth s gravity plays a key role in basketball and most other sports Acceleration Due to Gravity acceleration due to gravity g the acceleration that occurs when an object is allowed to fall freely close to Earth s surface g has a value of m s Acceleration due to gravity is the acceleration that occurs when an object is allowed to free fall the acceleration due to gravity of an object in the absence of air resistance fall freely The symbol for acceleration due to gravity is g Physicists have determined that the average value of g measured very close to Earth s surface is m s Different places on Earth have different values for g For example the value of g in Mexico City is slightly but measurably lower than the average of m s because the city has a very high elevation and is therefore farther from Earth s centre This had an interesting effect on the Summer Olympic Games which were held in Mexico City Many high jump long jump and pole-vaulting records were broken during these Olympic Games attributed in part to the lower value of g The value of g is different on different planets and other celestial objects The acceleration due to gravity of an object near Earth s surface will be about m s only if it is dropped in a vacuum This type of motion is referred to as free fall which is acceleration that occurs when there is no air resistance or other force affecting the motion of the object besides gravity Uniform Vertical Acceleration All objects that move freely in the vertical direction experience acceleration due to gravity g In the previous section you worked with the five key motion equations Since we know the average value of g close to Earth s surface we can use the motion equations and this knowledge to explore how gravity affects objects that are moving vertically Tutorial Motion of an Object Falling Straight Down The following Sample Problems will demonstrate using the five key equations of motion how to solve problems involving vertical motion You might want to review the equations in Table on page C -F -OP USB ai Sample Problem Determining the Time It Takes for an Object to Fall to the Ground A flowerpot is knocked off a window ledge and accelerates Solution uniformly to the ground If the window ledge is m above the ground and there is no air resistance how long does it take the flowerpot to reach the ground Chapter Motion in a Straight Line We know that the motion of the flowerpot is straight down However we cannot describe the direction of the vector as we have previously using E W N or S So we will let vectors NEL ---- SPLITTER -- with directions up and to the right be indicated by positive values Vectors with directions down and to the left will be indicated by negative values Since the flowerpot is not thrown upward or downward we can assume that it was initially at rest Therefore the initial velocity is m s Given Dd m a g m s v i m s Required t Analysis Since we are given the displacement acceleration and initial velocity of the flowerpot we can use Equation to solve for time Dd v iDt a Dt Solution Notice that the given displacement and acceleration values are both negative because both vectors point downward In general it is not valid to divide one vector by another However since the motion is in a straight line and the directions are given by the minus signs we are able to divide the vector values This equation can be rewritten as follows Dd a Dt Dt Dt m m s Dd a Dt s We know we should take the positive root because time intervals are always positive Statement The flowerpot will take s to reach the ground Sample Problem Determining the Final Velocity for a Falling Object What is the final velocity of the flowerpot in Sample Problem just before it hits the ground that only requires you to use given information We will use the vector directions for the given variables but will only calculate the magnitude of the final velocity We know that the direction of the final velocity must be downward vf aDd m s vf vi aDd vf vi aDd Solution To solve this problem we need to find the final velocity of the flowerpot just before it hits the ground If the question asked for its velocity after it had hit the ground and came to rest then the final velocity would be m s Here we are determining the velocity while the flowerpot is still in motion just before it hits the ground We will again write vectors with directions that are up and to the right as having positive values and those with directions down and to the left as having negative values Given Dd m a g m s v i m s Required v f Analysis We can choose any of the five key equations of motion that has v f as a variable It is always wise to choose the equation Solution Since the initial velocity is zero a m b m s Statement The flowerpot is travelling at m s downward just before it hits the ground Practice A ball is dropped from the roof of a building If it takes the ball s to reach the ground how tall is the building T I ans m A hot air balloon is hovering at height of m above the ground A penny is dropped from the balloon Assume no air resistance T I a How long does it take the penny to hit the ground ans s b What is the final velocity of the penny just before it hits the ground ans m s Tutorial Motion of an Object Thrown Straight Up The following Sample Problems involve analyzing the motion of an object that is first thrown straight up and then falls to Earth Sample Problem Determining the Height Reached by a Ball Thrown Straight Up in the Air A tennis ball is thrown straight up in the air leaving the person s hand with an initial velocity of m s as shown NEL in Figure on the next page How high from where it was thrown does the ball go Acceleration Near Earth s Surface ---- SPLITTER -- vf upward vector will be made positive and the downward vector negative We will consider the motion of the ball as it goes upward to its maximum height At its maximum height the ball comes to rest momentarily so the final velocity v f is zero Then we can use Equation Given v i m s a m s v f m s Required Dd Analysis vf vi aDd vi aDd vi aDd a vi Dd Solution Dd Figure The motion of a ball thrown straight upward Figure shows the path of the tennis ball Notice that the vectors for the initial velocity v i and the acceleration a are in opposite directions to each other Using our previous convention the m s m s Dd m Statement The maximum height attained by the tennis ball is m cm m b s m a b s vi a a vi a Sample Problem Determining the Time for a Ball Thrown Upward to Attain Its Maximum Height How long will it take the ball shown in Figure to reach its maximum height Given v i m s a m s v f m s Required t Analysis To determine the time that this motion takes we will identify an equation that uses only given information vf vi a Dt vf vi Dt a Since the final velocity is equal to zero v i Dt a v i Solution Dt a m a b s m s Dt s Statement It will take the tennis ball s to reach its maximum height Practice A golf ball is thrown straight up in the air at a velocity of m s T I a Determine the maximum height of the golf ball ans m b How long will it take the ball to reach its maximum height ans s c How long will it take the ball to fall from its maximum height to the height from which it was initially launched ans s A rock is thrown downward from a bridge that is m above a small creek The rock has an initial velocity of m s downward What is the velocity of the rock just before it hits the water T I ans m s down Chapter Motion in a Straight Line NEL ---- SPLITTER -- C -P -OP USB Free Fall and Terminal Velocity In real-life situations there will always be some air resistance Sometimes air resistance can be enough to have a significant effect on the motion of a falling object For example when a parachutist jumps out of an aircraft he can control the amount of air resistance based on how he positions his body If the parachutist dives out of the aircraft head first he will experience very little air resistance Most parachutists will try to fall so that as much of the surface area of their body is in contact with the air as possible In other words most parachutists will fall in a belly flop Figure When the air resistance on the parachutist is equal to the force due to gravity acting on the parachutist the parachutist will stop accelerating and stay at a constant velocity called the terminal velocity You will learn more about air resistance and free fall in Unit Figure The value of m s for acceleration near Earth assumes that there are no other forces acting on an object such as air resistance Summary terminal velocity the velocity of an object when the force due to air resistance equals the force due to gravity on the object The symbol g is used to represent the acceleration due to gravity All objects in free fall close to Earth s surface will accelerate at m s toward Earth s centre Air resistance can cause objects to accelerate at values less than g When an object reaches terminal velocity it will fall at a constant velocity Questions Describe the motion of an object that is dropped close to Earth s surface K U C A basketball player jumps up to make a basket and appears to hang in mid-air Write a brief description explaining to a Grade student what is occurring and why C A baseball is thrown straight up in the air reaches its maximum height and falls back down to the height from which it was originally thrown What is the acceleration of the ball a halfway up to its maximum height b at its maximum height c halfway back down to the initial height from which it was thrown T I A rubber ball is dropped from a height of m T I a How long does it take to hit the ground b What is the velocity of the ball when it has travelled a distance halfway to the ground An arrow is shot straight up into the air at m s T I a What is the arrow s maximum height b How long does the arrow take to reach its maximum height c Determine the total amount of time that the arrow is in the air A rock is thrown down from the top of a cliff with a velocity of m s down The cliff is m above the ground Determine the velocity of the rock just before it hits the ground T I Describe the motion of the object represented by the velocity time graph in Figure Give an example of an object that might undergo this type of motion K U C -F -OP USB v m s up Figure Velocity v Time for Accelerated Motion t s Research and describe a real-life situation where an object or person experiences an acceleration greater A than g m s gO TO NELSON SCiENCE NEL Acceleration Near Earth s Surface ---- SPLITTER -- Defining the Issue Researching Identifying Alternatives Explore an Issue in Vehicle Safety SKILLS MEnU Analyzing Defending a Decision Communicating Evaluating Electronic Speed Limiters for Teen Drivers Today s automobiles are the safest vehicles ever Advances in automotive technology have produced many safety devices including seat belts airbags electronic stability control electronic brake force distribution and antilock brakes All of these technologies are known to save lives when used correctly Transport Canada indicates that since the annual number of automobile accident fatalities in Canada has dropped by If we are to decrease the number of automobile accidents even further drivers will need to improve their safe driving practices Statistics from the U S National Highway Traffic Safety Administration NHTSA show that teenage drivers are more likely to take driving risks such as speeding than older drivers Speeding is a contributing factor in of all fatal crashes Statistics also show that teens are less likely to wear seat belts than other drivers In addition to contributing to driving accidents speed can affect the emissions released by vehicles as well as their efficiency Automobile emissions make up a substantial portion of greenhouse gas emissions Vehicle manufacturers have embraced advances in automobile technology to produce substantially more fuel-efficient vehicles Even a highly fuel-efficient vehicle is at its most efficient when driven at a reduced speed Transport Canada indicates that by reducing a vehicle s speed from km h to km h fuel consumption could drop by as much as In Ontario and Qu bec legislation has recently been passed to limit the speed of transport trucks to km h Figure A speed limiter is an electronic device that prevents a vehicle from travelling above a certain speed This has been suggested as a way to increase highway safety and reduce the environmental impact of truck emissions Some opponents of speed limiters say that the inability to go over a particular speed may prevent drivers from avoiding accidents Others say that there is not enough evidence to prove that speed limiters actually prevent accidents and that speed limiters can be tampered with C -P -OP USB Figure Many truck drivers object to the use of devices to limit speeds arguing that they will cause more frustration on the road The Issue Suppose you are a member of the provincial student council association The provincial government has tabled a bill that will call for the implementation of mandatory speed limiters for teenage drivers The government is proposing that all automobiles sold in Ontario have an electronic device installed that will electronically limit teen drivers to a maximum speed of km h sound an alarm when the vehicle reaches speeds of km h km h and km h The provincial government has argued that installing these devices will increase highway safety reduce automobile emissions and protect new drivers The provincial student council association will be making a presentation during a town hall meeting in which the proposed legislation is being discussed Members of the Ontario Ministry of Transportation will be present Goal To convince the Ministry of Transportation to either support this bill in its current format recommend modifications to the bill not support the bill Chapter Motion in a Straight Line NEL ---- SPLITTER -- ROLE You will be acting as a member of the provincial student council association representing young drivers in Ontario AUDIEnCE Your audience will be the Ministry of Transportation but you will present your views at a town hall meeting where the speed limiter legislation is being discussed Research SKILLS HANDBOOK A Conduct library and Internet research about speed limiters and their effectiveness Collect information about teen driving habits and their effect on road safety You might conduct a survey among your fellow students for their opinions You may wish to research the following What positive e ects have speed limiters had on safety among truck drivers What negative e ects have speed limiters had on safety Would those who want to break the law be able to find a way to disable a speed limiter WEB LInK To learn more about speed limiters gO TO NELSON SCiENCE Identify Solutions After reviewing your research consider the impact the speed limiter bill could have on you and on the citizens of Ontario What are the benefits and drawbacks of using speed limiters and whom would they harm or benefit What modifications to the bill might you recommend Make a Decision Based on your research decide which of the three possible solutions you will support Alternatively come up with a solution of your own Communicate Prepare a position paper photo essay or other presentation format supporting your chosen decision Be sure to support your decisions with research that will convince your audience the Ministry of Transportation to support your choice Plan for Action This issue has the potential to affect you and your classmates for many years Make a plan to make students aware of the potential for speed limiters that will electronically control driving behaviour among young drivers Your plan should include an opportunity for young drivers to voice their opinions on this issue For example you might plan to start a group on a social networking site or write an article for your school paper NEL Explore an issue in Vehicle Safety ---- SPLITTER -- CHAPTER Investigations OBSERVATIONAL STUDY Questioning Researching Hypothesizing Predicting Planning ontrolling C Variables Performing SKILLS MEnU Investigation watch Your Speed In this observational study you will use very simple equipment to measure the speed of real-life objects Observing Analyzing Evaluating Communicating purpose SKILLS HANDBOOK A In this study you will use a stopwatch and a tape measure to determine the average speed of vehicles driving past your school You will then determine the percentage of vehicles exceeding the posted speed limit Equipment and Materials m m tape measure stake or other marker stopwatch Start your stopwatch as each vehicle passes the starting point and stop the stopwatch as the vehicle passes the stake representing the end of your measured distance Record your observations Use the defining equation for average speed Dd vav to determine the average speed of Dt the vehicle Repeat this procedure for vehicles Determine the speed limit in front of your school Calculate the percentage of vehicles passing your school that are exceeding the posted speed limit procedure Choose a location near your school where vehicles will be travelling at what appears to be a constant speed For example be sure not to be close to a stop sign or stoplight Choose an obvious landmark such as a telephone pole as a