Physics Solution Manual for 1100 and 2101

# By combining these two relations we see that the

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Unformatted text preview: n vTG = –55 m/s and vGC = –5.0 m/s, the velocity of the truck relative to the car is vTC = vTG + vGC −55 m/s − 5.0 m/s = = − 56.3 m/s vTG vGC ( −55 m/s )( −5.0 m/s ) 1+ 1+ 2 c ( 65 m/s )2 The speed of the truck relative to the car is the magnitude of this result, or 56.3 m/s . ______________________________________________________________________________ 37. REASONING The velocity of the Enterprise 2, as measured by an earth-based observer, is given by v2e = v21 + v1e vv 1+ 2121e c (28.8) Chapter 28 Problems 1501 where v2e = velocity of Enterprise 2 relative to earth v21 = velocity of Enterprise 2 relative to Enterprise 1 v1e = velocity of Enterprise 1 relative to earth All of these variables are known, so v2e can be determined. SOLUTION The velocity of Enterprise 2 relative to the earth is ( +0.31c ) + ( +0.65c ) v21 + v1e = = +0.80c ( +0.31c )( +0.65c ) v21v1e 1+ 1+ 2 c2 c ______________________________________________________________________________ v2e = 38. REASONING We define the following relative velocities, assuming that the rocket approaching the earth from the right is traveling in the positive direction: vRL = velocity of the Right rocket relative to the Left rocket vRE = velocity of the Right rocket relative to the person on Earth = +0.75c vLE = velocity of the Left rocket relative to the person on Earth = –0.65c The velocity vRL can be found from the velocity-addition formula, Equation 28.8: vRL = vRE + vEL vv 1 + RE 2 EL c The velocity vRE is given, but vEL, the velocity of the earth relative to the left rocket, is not. However, we know that vEL is the negative of vLE, so vEL = – vLE = – (–0.65c) = +0.65c. SOLUTION The velocity of the right rocket relative to the left rocket is vRL = vRE + vEL +0.75c + 0.65c = = + 0.94c vRE vEL ( +0.75c )( +0.65c ) 1+ 1+ 2 c2 c The relative speed between the two rockets is the magnitude of this result, or 0.94c . ______________________________________________________________________________ 1502 SPECIAL RELATIVITY 39. SSM WWW REASONING Since the crew is initially at rest relative to the escape pod, the length of 45 m is the proper length L0 of the pod. The length of the escape pod as determined by an observer on earth can be obtained from the relation for length contraction ( ) 2 given by Equation 28.2, L = L0 1 – vPE / c 2 . The quantity vPE is the speed of the escape pod relative to the earth, which can be found from the velocity-addition formula, Equation 28.8. The following are the relative velocities, assuming that the direction away from the earth is the positive direction: vPE = velocity of the escape Pod relative to Earth. vPR = velocity of escape Pod relative to the Rocket = –0.55c. This velocity is negative because the rocket is moving away from the earth (in the positive direction), and the escape pod is moving in an opposite direction (the negative direction) relative to the rocket. vRE = velocity of Rocket relative to Earth = +0.75c These velocities are related by the velocity-addition formula, Equation 28.8. SOLUTION The relative velocity of the escape pod relative to the earth is vPE = vPR + vRE –0.55c + 0.75c = = + 0.34c vPR vRE (–0.55c )( +0.75c) 1+ 1+ c2 c2 The speed of the pod relative to the earth is the magnitude of this result, or 0.34c. The length of the pod as determined by an observer on earth is L = L0 1 − 2 vPE = (45 m) 1 – ( 0.34c )2 = 42 m 2 2 c c ______________________________________________________________________________ 40. REASONING The passengers would measure a length for the spaceships that does not match the constructed length if the two ships have a nonzero relative speed. This would mean that the ships are moving with respect to one another. If so, the phenomenon of length contraction would occur, and the passengers in either ship would measure a contracted length. To calculate the contracted length L, the length contraction formula must be used, as given in Equation 28.2: L = L0 1 − 2 vAB c2 (28.2) Chapter 28 Problems 1503 where L0 is the proper length of the spaceships, that is, the length measured by an observer at rest with respect to them. In other words, the proper length is the constructed length. The velocity vAB in Equation 28.2 is the velocity of spaceship A with respect to spaceship B. It can be obtained from the velocities of each ship with respect to the earth by using the velocity addition equation, as given in Equation 28.8: vAB = vAE + vEB vv 1 + AE 2 EB c (28.8) where vAE is the velocity of spaceship A with respect to the earth and vEB is the velocity of the earth with respect to spaceship B. We note that vEB = −vBE, where vBE is the velocity of spaceship B with respect to the earth. Substituting vEB = −vBE into Equation 28.8 gives vAB = vAE − vBE vv 1 − AE 2 BE c (1) SOLUTION Using Equation (1) to calculate vAB for use in Equation 28.2, we find vAB = vAE − vBE 0.850c − 0.500c = = 0.609c vAE vBE ( 0.850c )( 0.500c ) 1− 1− c2 c2 Here we have taken the direction in which the spaceships are traveling to be the positive direction. Substituting this result into Equation 28.2 reveals that...
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