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Conceptual Exercises 1) State the two fundamental postulates of Einstein's Special Relativity. Answer: (1) the constancy of the speed of light; (2) the laws of nature are the same in all inertial frames. Diff: 1 Page Ref: Sec. 26.2 2) State the Principle of Equivalence. Answer: An inertial reference frame in a uniform gravitational field is physically equivalent to a reference frame that is not in a gravitational field, but that is in uniform linear acceleration. Diff: 1 Page Ref: Sec. 26.5 3) Jennifer is inside her windowless space station laboratory and finds herself increasingly pulled to the floor. What experiment can she perform inside the lab to decide if she is feeling the effect of an acceleration or an effect of gravity? Answer: None. The Equivalence Principle says no experiment performed in a closed system can distinguish between the two effects. Diff: 3 Page Ref: Sec. 26.5 4) Using the Equivalence Principle as a guide, explain how one might cancel out (or make disappear) a gravitational field. Answer: If the observer accelerates in the direction of the gravitational field with the "acceleration of gravity" then all local gravitational effects will apparently disappear. Diff: 3 Page Ref: Sec. 26.5 5) The laws of physics are the same in all inertial frames of reference. Answer: True False Diff: 1 Page Ref: Sec. 26.2 6) Moving clocks are observed to run more slowly than clocks that are at rest in the observer's own frame fo reference. Answer: True False Diff: 1 Page Ref: Sec. 26.3 7) An object's length is largest when measured by an observer at rest with respect to the object. Answer: True False Diff: 1 Page Ref: Sec. 26.3 8) According to Einstein's general theory of relativity, rotating reference frames cannot be distinguished from gravitational acceleration. Answer: True False Diff: 1 Page Ref: Sec. 26.5 9) According to Einstein's theory of general relativity, a ray of light bends in a gravitational field. Answer: True False Diff: 1 Page Ref: Sec. 26.5 10) Albert Michelson is primarily remembered for A) proving time dilation. B) proving length contraction. C) proving that the speed of light is not constant. D) his theory published in 1916. E) not being able to detect an "ether". Answer: E Diff: 1 Page Ref: Sec. 26.1 11) The Michelson-Morley experiment was designed to measure A) the relativistic mass of the electron. B) the velocity of the Earth relative to the ether. C) the relativistic energy of the electron. D) the acceleration of gravity on the Earth's surface. Answer: B Diff: 1 Page Ref: Sec. 26.1 12) Michelson and Morley concluded from the results of their experiment that A) the experiment was successful in detecting a shift in the interference pattern. B) the experiment was a failure since they detected a shift in the interference pattern. C) the result was a surprise since there was no detectable shift in the interference pattern. Answer: C Diff: 1 Page Ref: Sec. 26.1 13) You can build an interferometer yourself if you use the following components: A) a light source, a detector screen, a partially silvered mirror, two flat mirrors, and a glass plate. B) a light source, a detector screen, a partially silvered mirror, a flat mirror, and a glass plate. C) a light source, a detector screen, two partially silvered mirrors, and a glass plate. D) a light source, a detector screen, two partially silvered mirrors, a flat mirror, and a glass plate. Answer: A Diff: 1 Page Ref: Sec. 26.1 14) One of Einstein's postulates in formulating the special theory of relativity was that the laws of physics are the same in reference frames that A) oscillate. B) are stationary, but not in moving frames. C) accelerate. D) move at constant velocity with respect to an inertial frame. Answer: D Diff: 1 Page Ref: Sec. 26.2 15) The theory of special relativity A) does not agree with Newtonian mechanics. B) is based on a complex mathematical analysis. C) does not agree with electromagnetic theory. D) has not been verified by experiment. Answer: A Diff: 1 Page Ref: Sec. 26.2 16) You are riding in a spaceship that has no windows or other means for you to observe or measure what is outside. You wish to determine if the ship is stopped or moving at constant velocity. What should you do? A) You can determine if the ship is moving by checking your precision time piece. If its running slow, the ship is moving. B) You should give up because you have taken on an impossible task. C) You can determine if the ship is moving by determining the apparent velocity of light. D) You can determine if the ship is moving either by determining the apparent velocity of light or by checking your precision time piece. If its running slow, the ship is moving. Answer: B Diff: 1 Page Ref: Sec. 26.2 17) One consequence of Einstein's theory of special relativity is that A) absolute velocity of stars can be measured with respect to the ether wind. B) absolute Earth velocity can be measured. C) simultaneity is not absolute. D) relative speed can not be measured. E) nothing is conserved because it changes. Answer: C Diff: 2 Page Ref: Sec. 26.2 18) In spite of the fictional exploits of the starship Enterprise, "Relativity" teaches us that the maximum possible speed is what? (hint: warp = v/c) A) "warp" 10 B) "warp" 1 C) "warp" D) "warp" infinity E) "warp" 0 Answer: B Diff: 2 Page Ref: Sec. 26.2 19) The gamma factor is 1 / 2 and it A) can be zero. B) can be any number greater than or equal to zero. C) can be any number greater than or equal to one. D) can be a negative number. E) cannot equal one. Answer: C Diff: 1 Page Ref: Sec. 26.3 20) An object moves in a direction parallel to its length with a velocity that approaches the velocity of light. The width of this object as measured by a stationary observer A) increases slightly. B) approaches infinity. C) approaches "gamma". D) does not change. E) approaches zero. Answer: D Diff: 1 Page Ref: Sec. 26.3 21) Relative to a stationary observer, a moving clock A) always runs slower than when at rest. B) can run faster or slower; it depends on the relative velocity between the observer and the clock. C) always runs faster than when at rest. D) keeps its normal time. Answer: A Diff: 1 Page Ref: Sec. 26.3 22) As the velocity of your spaceship increases, you would observe A) that your mass has increased. B) that the length of your spaceship has decreased. C) that your precision clock runs slower than normal. D) none of the other answers. Answer: D Diff: 1 Page Ref: Sec. 26.3 23) An object moves in a direction parallel to its length with a velocity that approaches the velocity of light. The length of this object as measured by a stationary observer A) approaches zero. B) approaches infinity. C) does not change. D) increases slightly. Answer: A Diff: 2 Page Ref: Sec. 26.3 24) A spear is thrown at you at a very high speed. As it passes, you measure its length as one-half its normal length. From this measurement, you conclude that the moving spear's mass must be A) twice its rest mass. B) four times its rest mass. C) zero. D) one-half its rest mass. Answer: A Diff: 2 Page Ref: Sec. 26.3 25) If you were to measure your pulse rate while in a spaceship moving away from the sun at a speed close to the speed of light, you would find that it was A) the same as it was here on Earth. B) much slower than normal. C) much faster than normal. Answer: A Diff: 2 Page Ref: Sec. 26.3 26) Consider a particle of mass m and rest mass mo. Which of the following is the correct expression for the kinetic energy of such a particle? A) moc2 + mov2/2 B) m v2/2 C) mc2 – moc2 D) 1/2(m c2 – moc2) E) mov2/2 Answer: C Diff: 1 Page Ref: Sec. 26.4 27) The total energy of a particle at rest is A) moc2. B) (1/2) moc2. C) zero. D) p2/(2mo) . Answer: A Diff: 1 Page Ref: Sec. 26.4 28) Which of the following depends on the observer's frame of reference? A) all of the other answer choices B) the length of a meter stick C) the mass of the proton D) the half-life of a muon Answer: A Diff: 1 Page Ref: Sec. 26.4 29) What happens to the kinetic energy of a speedy proton when its relativistic mass doubles? A) It must increase, but it is impossible to say by how much. B) It less than doubles. C) It more than doubles. D) It doubles. Answer: C Diff: 2 Page Ref: Sec. 26.4 30) George Gamow, the creator of the Big-Bang theory of the origin of the universe, asks you to imagine what would happen if the speed of light was 15. mi/h. If you were on the sidewalk looking at a bicyclist ride down the street, how does she look to you while riding, compared to when she stops? A) taller, the same mass, but stretched out like a limousine B) same height, greater mass, and flatter in the direction she is moving C) shorter, greater mass, but stretched out like a limousine D) same height, smaller mass, and flatter in the direction she is moving E) shorter, the same mass, but stretched out like a limousine Answer: B Diff: 2 Page Ref: Sec. 26.4 31) What happens to the total relativistic energy of a speedy proton when its relativistic mass doubles? A) It doubles. B) It less than doubles. C) It must increase, but it is impossible to say by how much. D) It more than doubles. Answer: A Diff: 2 Page Ref: Sec. 26.4 32) An object moves in a direction parallel to its length with a velocity that approaches the velocity of light. The mass of this object as measured by a stationary observer A) does not change. B) increases slightly. C) approaches zero. D) approaches infinity. Answer: D Diff: 2 Page Ref: Sec. 26.4 33) Compared to "Special" Relativity, "General" Relativity is more concerned with A) electromagnetic fields. B) gravitation. C) Unified fields. D) mass-energy. E) Lorentz transformations. Answer: B Diff: 1 Page Ref: Sec. 26.5 34) Black holes A) are the collapsed remnant of stars. B) cannot be detected in binary star systems. C) are holes in space, devoid of matter. D) are predicted by Einstein's special theory of relativity. Answer: A Diff: 1 Page Ref: Sec. 26.5 35) When star light passes by the sun A) it is unaffected by the sun. B) it is absorbed by the sun. C) it is deflected away from the sun. D) it is deflected toward the sun. Answer: D Diff: 2 Page Ref: Sec. 26.5 36) The gravitational red shift is caused by A) Rayleigh scattering in the atmosphere. B) rotating black holes. C) time dilation. D) gravitational lensing. Answer: C Diff: 2 Page Ref: Sec. 26.5 37) The Schwarzchild radius of a black hole is that radial distance from the center of a sphere within which not even light can escape. It was first discovered mathematically by Schwarzchild in 1916 after Einstein published his general relativity theory. It can be calculated from a star's mass M as: R = 2GM/c2. If the mass of star G is twice as much as the mass of star H, the average density of star G compared to star H will be A) twice as much. B) four times as much. C) half as much. D) one-fourth as much. Answer: D Diff: 3 Page Ref: Sec. 26.5 38) Consider two spaceships, each traveling at 0.500c in a straight line. Ship A is moving directly away from the sun and ship B is approaching the sun. The science officers on each ship measure the velocity of light coming from the sun. What do they measure for this velocity? A) On both ships it is measured to be greater than c. B) Ship B measures it as less than c, and ship A measures it as greater than c. C) On both ships it is measured to be less than c. D) On both ships it is measured to be exactly c. E) Ship A measures it as less than c, and ship B measures it as greater than c. Answer: D Diff: 1 Page Ref: Sec. 26.6 Quantitative Exercises 1) A spaceship visits Alpha Centauri and returns to Earth. Alpha Centauri is 4.5 light-years from Earth (our second closest star). If the spaceship travels at one-half the speed of light for essentially all of its expedition, how long was the ship gone according to an observer on the Earth? Answer: 10. light-years Diff: 2 Page Ref: Sec. 26.3 2) At what speed are lengths contracted to half and times dilated by a factor of 2? Answer: 0.866 c Diff: 2 Page Ref: Sec. 26.3 3) How fast should a moving clock travel if it is to be observed by a stationary observer as running at one-half its normal rate? Answer: 0.866c Diff: 2 Page Ref: Sec. 26.3 4) A muon at rest decays in 2.2 ?s. Moving at 99.% the speed of light, it would be seen to "live" for how long? Answer: 16. ?s Diff: 2 Page Ref: Sec. 26.3 5) A spaceship visits Alpha Centauri and returns to Earth. Alpha Centauri is 4.5 light-years from Earth (our second closest star). If the spaceship travels at one-half the speed of light for essentially all of its expedition, how long was the ship gone according to an observer on the spaceship? Answer: 7.8 years Diff: 2 Page Ref: Sec. 26.3 6) How fast would a rocket ship have to move to contract to half of its proper length (as observed by a stationary object)? Answer: 0.87 c Diff: 2 Page Ref: Sec. 26.3 7) You are moving past the Earth at 0.99c and notice your heart beating 88. times/minute. (a) Your doctor on Earth observes your heart rate to be what? (b) If you had been moving 99.99% the speed of light, you would have observed your heart to be beating how fast? Answer: (a) 12. beats/min (b) 88. beats/min Diff: 3 Page Ref: Sec. 26.3 Skill: Algorithmic 8) Aliens on Krypton build a spacecraft 472. teef long. In their units the speed of light is 22.2 million teef/ces. How fast (teef/ces) must it travel to appear to be 66.6 teefs long? Answer: 0.99c = 22. × 106 teef/ces Diff: 3 Page Ref: Sec. 26.3 9) A stretch of land is 12.5 km long on a map. (a) What would be its length to an observer moving by, parallel to the stretch, at 0.95c? (b) Which length is the proper length? (c) How fast must the observer move for the land to appear 6.50 km long? Answer: (a) 3.9 km (b) 12.5 km (c) 0.854c Diff: 3 Page Ref: Sec. 26.3 Skill: Algorithmic 10) A starship is built to plans that state it is to be constructed 100. meters long. It is launched and as it coasts by the Earth, it is seen to be shortened to 5.48 meters long. (a) How fast is it moving? (b) The ship's crew measure the length of the ship to be how long? Answer: (a) 99.85% the speed of light (b) 100. meters Diff: 3 Page Ref: Sec. 26.3 11) Consider a student whose mass is 70. kg (he weighs 154. lb). If all his mass were converted to energy, this energy could light a 100. watt lamp for how long? Answer: 6.3 × 1016 s = 2.0 billion years! Diff: 2 Page Ref: Sec. 26.4 12) How fast must something be traveling if its mass increases by 10%? Answer: 1.25 × 108 m/s Diff: 2 Page Ref: Sec. 26.4 13) An electron has a relativistic momentum of 1.1 × 10-21 kg-m/s. What fraction of its total energy is its kinetic energy? Answer: 0.75 Diff: 2 Page Ref: Sec. 26.4 14) The atomic bomb that was dropped on Nagasaki in 1945 killed 140,000 people, helping to end World War II on the next day. It released energy equivalent to that of 20,000 tons of TNT explosive. How much mass was converted to energy when this took place? (1000 tons ? 4.3 × 1012 J) (Incidentally, modern H-bombs have energy yields 1000 times as much!) Answer: 9.6 × 10-4 kg Diff: 2 Page Ref: Sec. 26.4 15) A person of initial mass 70. kg climbs a stairway, rising 6.0 m in elevation. By how much does his mass increase by virtue of his increased potential energy? Answer: 4.6 × 10-14 kg Diff: 2 Page Ref: Sec. 26.4 16) If the mass of a 1.0 kg book could be entirely converted into electrical energy, how many Kw-h would that generate? Answer: 2.5 × 1010 Kw-h ( = 9.0 × 1016 J) Diff: 3 Page Ref: Sec. 26.4 17) An alpha particle (6.70 × 10-27 kg, 3.2 × 10-19 Coulomb) travels at 0.965c. What is its: (a) rest energy (in eV)? (b) total energy (in eV)? (c) kinetic energy (in eV)? Answer: (a) 3.76 GeV (b) 14.4 GeV (c) 10.6 GeV Diff: 3 Page Ref: Sec. 26.4 Skill: Algorithmic 18) Above what speed do the Relativistic and the Newtonian expressions for KINETIC ENERGY disagree by more than 1%? Answer: v/c = 0.12 (v = 3.6 × 107 m/s ) Diff: 3 Page Ref: Sec. 26.4 19) An electron slowing from 0.998c to 0.500c loses how much momentum? (express the answer in MeV/c as well as N-s) Answer: 7.77 MeV/c = 4.15 × 10-21 N-s Diff: 3 Page Ref: Sec. 26.4 20) Suppose a 70. kg student were to become (be compressed into) a black hole. (a) What would be his/her Schwarzshild radius ("size" of the B.H.)? (b) How does this compare to the size of a proton? Answer: (a) 1.0 × 10-25 meters (b) About 10 orders of magnitude smaller than a proton. Diff: 3 Page Ref: Sec. 26.5 21) A person in a rocket ship traveling past the earth at a speed of 0.500c fires a laser gun in the forward direction. With what speed does an observer on Earth see the light pulse travel? Answer: c = 3.00 × 108 m/s Diff: 1 Page Ref: Sec. 26.6 22) From the Earth, they see the Enterprise approaching at 0.800c and the Klingons approaching from the opposite direction at 0.900c. From the Enterprise, the crew sees the Klingon ship approaching at what speed? Answer: Klingons approaching the Enterprise at 0.988c Diff: 2 Page Ref: Sec. 26.6 23) Two spaceships are traveling through space at velocities of 0.600c and 0.900c, respectively, with respect to Earth. If they are headed directly toward each other, what is their approach velocity, as measured by the captain of either ship? Answer: 0.974c Diff: 2 Page Ref: Sec. 26.6 24) Two spaceships approach each other along a straight line at a constant velocity of 0.988c as measured by the captain of one of the ships. An observer on Earth is able to measure the speed of only one of the ships as 0.900c. From the point of view of the observer on Earth, what is the speed of the other ship? Answer: 0.794c Diff: 2 Page Ref: Sec. 26.6 25) A radar operator on Earth sees two spaceships moving straight at each other, each with speed 0.60c. With what speed does the pilot of one ship see the other ship approaching? Answer: 0.88c Diff: 2 Page Ref: Sec. 26.6 26) A person onboard spaceship A sees craft B moving away at 92.2% the speed of light and he sees C approaching at 0.833c (see Figure 26-1). Figure 26-1 (a) Spacecraft B sees C approaching with what speed? (b) Ship C sees B approaching with what speed? (c) Ship C sees A approaching at what speed? (d) Spacecraft B determines A to be moving at what speed? Answer: (a) approaches at 0.993c (b) approaches at 0.933c (c) approaches at 0.833c (d) recedes at 0.922c Diff: 3 Page Ref: Sec. 26.6 27) Using the velocity transformation formula, calculate the speed in the "rest frame" of a particle which is seen to be moving at c in a "moving frame". Answer: u = (v+u')/(1+vu"/c2) = (v+c)/(1+vc/c2) = c(v+c)/(c+v) = c Light speed in one frame will be light speed in all inertial frames. Diff: 3 Page Ref: Sec. 26.6 28) Earth observes a rocket move away at 0.370c. The rocket is designed to launch a projectile at 0.505c relative to the rocket. (a) Fired straight ahead, what is the projectile speed observed from Earth? (b) Launched to the rear, what is the projectile speed observed from Earth? Answer: (a) 0.737c (b) 0.166c Diff: 3 Page Ref: Sec. 26.6 Skill: Algorithmic 29) At what speed would a 100. meter long spaceship appear to be 60. meters long? A) 24. Mm/s B) 0.24 m/s C) 2.4 × 103 Km/s D) 2.4 Gm/s E) 240. Mm/s Answer: E Diff: 2 Page Ref: Sec. 26.3 30) How fast must a spaceship travel to make the trip to a star 400. LY away in 25.3 years of astronaut travel time? A) 0.966c B) 0.987c C) 0.829c D) 0.998c E) 0.992c Answer: D Diff: 2 Page Ref: Sec. 26.3 31) Suppose one observes cosmic rays creating a new particle high in the atmosphere and the particle moves at 99.7% the speed of light. It is observed to decay in an average 37.0 ?s. What is the proper lifetime of the particle? A) 2.86 ?s B) 18.1 ?s C) 5.33 ms D) 12.4 ?s E) 4.33 ?s Answer: A Diff: 2 Page Ref: Sec. 26.3 32) What speed dilates 1.000 second into 70.71 seconds? A) 0.9999c B) 0.8777c C) 0.9777c D) 0.9888c E) 0.9666c Answer: A Diff: 2 Page Ref: Sec. 26.3 33) A stick is moving along the x-axis at 90%c. A person riding with the stick sees it inclined at 45.° with respect to the x-axis. Observers in the rest frame see the stick making what angle with respect to the x-axis? A) 53.° B) 66.° C) 88.° D) 45.° E) 79.° Answer: B Diff: 2 Page Ref: Sec. 26.3 34) The amount of energy equivalent to one kilogram of mass at rest is A) 1.2 × 108 eV. B) 9.0 × 1016 J. C) 4.5 × 1016 J. D) 6.0 × 10-3 eV. E) 3.0 × 108 J. Answer: B Diff: 2 Page Ref: Sec. 26.4 35) If one uses the classical expression for kinetic energy (1/2 m v2) for a particle traveling at half the speed of light, the result will be deficient by how many percent? A) 24% B) 13% C) 27% D) 7% E) 1% Answer: A Diff: 2 Page Ref: Sec. 26.4 36) An electron is accelerated through 100 kV. By what factor has its mass increased with respect to its rest mass? A) 8.00 B) 4.25 C) D) 1.55 E) 1.20 Answer: E Diff: 2 Page Ref: Sec. 26.4 37) A proton (1.67 × 10-27 kg) is accelerated to 99.8% the speed of light. How much energy did the accelerator have to impart? A) 38.9 GeV B) 7.52 GeV C) 13.9 GeV D) 14.8 GeV E) 223. GeV Answer: C Diff: 2 Page Ref: Sec. 26.4 38) How much energy would be required to accelerate a 77. kg professor from rest to 90% the speed of light? A) 2.3 mc2 B) 1.3 mc2 C) 0.30 mc2 D) 0.50 mc2 E) 0.41 mc2 Answer: B Diff: 2 Page Ref: Sec. 26.4 39) How many joules of energy are required to accelerate one kilogram of mass from rest to a velocity of 0.886c? A) 3.0 × 103 J B) 8.1 × 1021 C) 1.8 × 1017 J D) 3.4 × 10-7 E) 9.0 × 1016 J Answer: E Diff: 2 Page Ref: Sec. 26.4 40) A spaceship leaves Earth at 70.0% the speed of light. How fast must the spacecraft launch a shuttlecraft with respect to the ship so that it appears, from Earth, to be moving 98.7% the speed of light away. A) 0.911c B) 0.972c C) 0.928c D) 0.987c E) 0.966c Answer: C Diff: 2 Page Ref: Sec. 26.6 41) Two spaceships are traveling through space at 0.6c relative to the Earth. If the ships are headed directly toward each other, what is their approach velocity, as measured by a person on either craft? A) 0 B) c C) 0.6c D) 1.2c E) none of the above Answer: E Diff: 2 Page Ref: Sec. 26.6 42) A spaceship, traveling at WARP 1/2 away from the Earth, launches a shuttle moving away from the ship at c/2 (c/2 seen from the ship). Classically one expects to see the shuttle moving at c from the Earth, but relativistically what speed do we expect seen from the Earth? A) 0.9c B) 0.6c C) 0.5c D) 0.8c E) 0.7c Answer: D Diff: 2 Page Ref: Sec. 26.6 The following select End of Chapter Exercises from College Physics,Sixth Edition, Chapter 26 are also available in TestGenerator on the Instructor Resource Center on CD-ROM: Exercises: 5, 9, 10, 16, 24, 25, 27, 30, 34, 35, 44, 47, 50, 57, 59, 66, 70, 72, 73 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 500
