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University of Colorado : UC
Uploaded: A month ago
Contributor: ryann
Category: Physics
Type: Lecture Notes
Rating: A (1)
Filename:   Lec04.pdf (1.22 MB)
Page Count: 19
Credit Cost: 2
Views: 16
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1D Kinematics & Vectors Announcements: Look at the new items on the webpage Attend your tutorial section tomorrow. Opened 2 more seats in room G2B60 for tutorials – can see Leigh Dodd if need help • Your 4 digit CAPA number changes every week. Use the correct one. Web page: 1 Summary of constant acceleration equations No displacement in this equation No final velocity in this equation No time in this equation No acceleration in this equation 2 Set frequency to BA Clicker question 1 Q. If you drop an object in the absence of air resistance, 2 it accelerates downward at 9.8 m/s. If you throw it downward, its downward acceleration after release is 2 A.  more than 9.8 m/s 2 B.  9.8 m/s 2 C.  less than 9.8 m/s D.  depends on the mass of the ball The initial velocity has nothing to do with it! After release, the only acceleration is 2 due to gravity which is 9.8 m/s 3 Set frequency to BA Clicker question 2 Q. A person standing at the edge of a cliff throws one ball straight up and another ball straight down at the same initial speed. Neglecting air resistance, the ball to hit the ground below the cliff with the greater speed is the one initially thrown A: upward. B: downward. C: neither—they both hit at the same speed If you throw a ball up with v0, it will go up, stop, come back down... and when it reaches you, it will have the SAME speed as it started with, just going the other way. That means that either ball has the same (downward) speed when it goes by you - the one you threw up just takes longer... 2 2 If you prefer a formula: vf = v0 + 2*a*(height). In this case, v0 is either + or -, but when you square it, you get the same result! vf (final speed) is going to be the 4 same either way. Example 1: Deer in the Headlights The driver of a car going 30 m/s on a straight road sees a deer in the road 75 m ahead. What acceleration is needed to stop before hitting the deer? 30 m/s First, draw a picture 75 m What do we know? Solution 1: Using we get acceleration: 5 Example problem 1 (continued) What do we know? 30 m/s 75 m Solution 2: From we get Then solve to get Acceleration is negative because we defined velocity as positive and car is slowing Sometimes give acceleration in g’s: 6 Set frequency to BA Clicker question 3 Q. On planet X, a cannon ball is fired Time Height Velocity straight upward. The position and (s) (m) (m/s) velocity of the ball at many times are 0 0 20 listed at the right. Note that we have 1 17.5 15 chosen up as the positive direction. 2 30 10 What is the acceleration due to 3 37.5 5 gravity on Planet X? 4 40 0 2 A.  –5 m/s 2 5 37.5 -5 B.  –10 m/s 2 6 30 -10 C.  –15 m/s 2 D.  –20 m/s 7 17.5 -15 2 E.  5 m/s 8 0 -20 7 Solutions to clicker question 3 Solution 1: Use for constant Time Height Velocity (s) (m) (m/s) acceleration to get: 0 0 20 Solution 2: Use and solve 1 17.5 15 for a: 2 30 10 Solution 3: Use with 3 37.5 5 x=40 m, x0=0 m, v0=20 m/s, t = 4 s to get 4 40 0 5 37.5 -5 Solution 4: Use with 6 30 -10 x=0 m, x0=40 m, v0=0 m/s, t = 4 s to get 7 17.5 -15 so 8 0 -20 8 Set frequency to BA Clicker question 4 A truck traveling at 50 km/hr (about 14 m/s) approaches a car stopped at a red light. When the truck is 100 m from the car, the light turns green and the car immediately begins to accelerate at 2 2.0 m/s to a final speed of 100 km/hr. Which graph represents this situation? car car x x x A B C D. None of car the graphs t t t truck truck truck Can't be A or B, because in both of those the truck (straight line, i.e. constant v) is at or AHEAD of the car at t=0, where the light turned green. But in the problem, the truck is 100 m behind the car...So it's either C or D. The question is... does the truck ever pass the car (do the curves cross) or not? Need to do this with math!... Consider the time when the car has first reached the same speed as the truck... When/where does that happen? For constant a, Δv= a*t, so the car reaches 14 m/s at t=Δv/a = (14 m/s -0) /(2 m/s^2 )= 7 seconds. At that time, x(car) = 1/2*a*t^2 = (1/2)*(2 m/s^2)*(7 sec)^2 = 49 m. At that same time, Δx(truck) = 14 m/s * 7 sec = 98 m. Since it started off 100 meters back, the truck is STILL behind the car at the instant they have the same speed. That means curve C is reasonable! 9 Set frequency to BA Clicker question 5 A ball is launched straight up with initial velocity vo. (Neglect air resistance) If the initial velocity vo is doubled, the time to reach the apex of the trajectory... A: doubles. B: increases by a factor of 4. C: Neither of these. D: Not enough information given. Constant accel says vf = v0 + a*time, and here, vf = 0 because we're at the top. So, v0 = -a * time, which means if you double v0, you double the time. 10 Set frequency to BA Clicker question 6 A ball is launched straight up with initial velocity vo. (Neglect air resistance) If the initial velocity vo is doubled, the maximum height of the ball…. A: doubles. B: increases by a factor of 4. C: Neither of these. D: Not enough information given. 2 2 Constant accel says vf = v0 + 2*a*height. Again, with vf = 0, 2 we have v0 = - 2*a*height. So, if you double v0, that "squared" means you quadruple the height! 11 2D & 3D Kinematics • Displacement, velocity, & acceleration are vectors and so have magnitude and direction (which is why we needed to remember signs in 1D) • We will mostly work in 2D for the next couple of weeks but everything generalizes to 3D Trigonometry refresher: SOHCAHTOA Hypotenuse Opposite Need to understand trig functions Adjacent (including inverse) on calculator and understand degrees and radians 12 Vector representations • Can represent as an arrow • Can represent as ordered list of numbers in a known coordinate system (giving the components) • Can represent as a sum of components times unit vectors – Unit vectors have magnitude 1 and point along an axis: 13 Vector representation y Vector is represented by arrow 3 x 4 The vector or A is written with an arrow or in bold The magnitude is written in normal font |A| Components are which, if necessary, can be found by trigonometry: 14 Graphical vector addition Want to find where and Rule is head to tail: Vector addition is commutative: Can multiply vector by scalar multiplying the length but leaving the direction unchanged except a negative scalar flips direction 180° Vector subtraction done by adding the negative: 15 Set frequency to BA Clicker question 7 Q. Three vectors, B are shown. Which A C answer represents the vector ? A S B C So (A) (B) (C) (D) (E) None of these. 16 Vector addition by components Split vectors into orthogonal components and add components individually. For and then 17 Vector addition by components (2) Use trigonometry to split vectors into orthogonal components if you are given magnitude and direction but angles can be tricky 3.2 Draw quick diagram and determine angle -6.5 Need to determine how measured angle relates to angle from +x axis. In this case  C = (7.2,154°) 18 € Last Slide • Don’t forget Tutorial material for your recititation tomorrow! 19

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