1. In this problem you will derive the formula for time dilation, one of the con-...

   

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1. In this problem you will derive the formula for time dilation, one of the con- sequences of Einstein's postulates. The pictured apparatus consists of a flashing light source, F, that is a distance h from a mirror, M. Consider three events: X = F emits flash of light, Y = flash reflects off M, and Z = flash returns to F. Two observers, Anna and Bob, both with reliable clocks, meas- ure the time interval between events X and Z. Use only Einstein's postulates, the definition of velocity and elementary geometry. Assume h has the same observed value in both Anna's and Bob's frame of reference. M (a) Anna is in a frame of reference in which the apparatus is stationary. Find the time interval, Ato, in this frame of reference, expressed in terms of h and c. [Note: If two events happen at the same location in a given reference frame, then the time between those two events in that frame is called the "proper time". Hence, the time interval Anna measures between X and Z is the proper time for those two events.] (b) Bob is in a frame of reference in which the apparatus is moving to the right at constant speed v. Show that the time interval, At, in Bob's reference frame is: At= 1-(v/c) [Hint: Draw a picture depicting all three events, and remember that the flash of light travels at c in both Anna's and Bob's frames, per Einstein's 2nd postulate.] 2. Anna pilots a flying vehicle traveling in a straight line at speed v relative to Bob. When the vehicle passes a first marker, P, Anna and Bob start their respective high-precision stopwatches. They stop them when the vehicle passes a second marker, Q. Anna's stopwatch reads exactly 10.0000000000000000000000 minutes. What does Bob's stopwatch read if the vehicle is a jet plane traveling at 300 m/s? [Note: Include enough digits to show a difference from Anna's stopwatch.] (b) What does Bob's stopwatch read if the vehicle is a spaceship traveling at 106 m/s? (c) What does Bob's stopwatch read if the vehicle is a spaceship traveling at 107 m/s? (d) How fast does the vehicle have to be traveling relative to Bob in order for his stop- watch to read exactly 11 minutes? 3. According to Albert on Earth, the uninhabited planet Flerba is 5 ly away. Gleef, an alien from planet Yop, is on a spaceship, passing Earth at 0.8c, along a straight line path toward Flerba. She is heading to Flerba to collect kricks. Unfortunately, Flerba explodes. According to Albert, this occurred 2 yr after Gleef passed Earth. (Albert, of course, has to wait a while for the light from the explosion to arrive, but he reaches his conclusion by "working backward.") Call the passing of Gleef's spaceship past Albert on Earth time zero for both. [Hints: Break it all down into events. Use the Lorentz transformation; or else, use time dilation, length contraction and Einstein's 2nd postu- late. Units: use 1 yr for time, use 1 ly = 1 light-year = cx (1 yr) for length.] (a) According to Gleef, how far is her spaceship from Flerba when it explodes. (b) At what time does it explode, according to Gleef? (c) Does Flerba explode before or after Gleef passes Albert? Explain. 4. Bob is watching Anna fly by in her new high-speed rocket plane, which Anna knows to be 60 m in length. As a greeting. Anna flips on two lights simultaneously, one at the front of the rocket plane and one at the tail. According to Bob, the lights come on at different times, 40 ns apart. (a) Which light comes on first, according to Bob? (b) How fast is the rocket plane moving past Bob? [Give your answer in terms of c.] 5. In the frame in which they are at rest, the number of muons at time is given by: N = Noe where No is the number at = 0 and r is the mean lifetime 2.2 us. Suppose muons are produced at height of 4.0 km above the ground and head toward the ground at 0.93c. (a) What is the mean lifetime of a muon in the reference frame of the ground? (b) What is the distance to the ground from the perspective of the muons? (c) What fraction of the muons will survive to reach the ground? (d) What fraction would reach the ground if classical mechanics were valid? 6. A pion is an elementary particle that, on average, disintegrates 2.610-8 s after its cre- ation in a frame at rest relative to the pion. An experimenter finds that pions created in a laboratory travel 13 m on average before disintegrating. How fast are the pions trav- eling through the lab? 1. In this problem you will derive the formula for time dilation, one of the con- sequences of Einstein's postulates. The pictured apparatus consists of a flashing light source, F, that is a distance h from a mirror, M. Consider three events: X = F emits flash of light, Y = flash reflects off M, and Z = flash returns to F. Two observers, Anna and Bob, both with reliable clocks, meas- ure the time interval between events X and Z. Use only Einstein's postulates, the definition of velocity and elementary geometry. Assume h has the same observed value in both Anna's and Bob's frame of reference. M (a) Anna is in a frame of reference in which the apparatus is stationary. Find the time interval, Ato, in this frame of reference, expressed in terms of h and c. [Note: If two events happen at the same location in a given reference frame, then the time between those two events in that frame is called the "proper time". Hence, the time interval Anna measures between X and Z is the proper time for those two events.] (b) Bob is in a frame of reference in which the apparatus is moving to the right at constant speed v. Show that the time interval, At, in Bob's reference frame is: At= 1-(v/c) [Hint: Draw a picture depicting all three events, and remember that the flash of light travels at c in both Anna's and Bob's frames, per Einstein's 2nd postulate.] 2. Anna pilots a flying vehicle traveling in a straight line at speed v relative to Bob. When the vehicle passes a first marker, P, Anna and Bob start their respective high-precision stopwatches. They stop them when the vehicle passes a second marker, Q. Anna's stopwatch reads exactly 10.0000000000000000000000 minutes. What does Bob's stopwatch read if the vehicle is a jet plane traveling at 300 m/s? [Note: Include enough digits to show a difference from Anna's stopwatch.] (b) What does Bob's stopwatch read if the vehicle is a spaceship traveling at 106 m/s? (c) What does Bob's stopwatch read if the vehicle is a spaceship traveling at 107 m/s? (d) How fast does the vehicle have to be traveling relative to Bob in order for his stop- watch to read exactly 11 minutes? 3. According to Albert on Earth, the uninhabited planet Flerba is 5 ly away. Gleef, an alien from planet Yop, is on a spaceship, passing Earth at 0.8c, along a straight line path toward Flerba. She is heading to Flerba to collect kricks. Unfortunately, Flerba explodes. According to Albert, this occurred 2 yr after Gleef passed Earth. (Albert, of course, has to wait a while for the light from the explosion to arrive, but he reaches his conclusion by "working backward.") Call the passing of Gleef's spaceship past Albert on Earth time zero for both. [Hints: Break it all down into events. Use the Lorentz transformation; or else, use time dilation, length contraction and Einstein's 2nd postu- late. Units: use 1 yr for time, use 1 ly = 1 light-year = cx (1 yr) for length.] (a) According to Gleef, how far is her spaceship from Flerba when it explodes. (b) At what time does it explode, according to Gleef? (c) Does Flerba explode before or after Gleef passes Albert? Explain. 4. Bob is watching Anna fly by in her new high-speed rocket plane, which Anna knows to be 60 m in length. As a greeting. Anna flips on two lights simultaneously, one at the front of the rocket plane and one at the tail. According to Bob, the lights come on at different times, 40 ns apart. (a) Which light comes on first, according to Bob? (b) How fast is the rocket plane moving past Bob? [Give your answer in terms of c.] 5. In the frame in which they are at rest, the number of muons at time is given by: N = Noe where No is the number at = 0 and r is the mean lifetime 2.2 us. Suppose muons are produced at height of 4.0 km above the ground and head toward the ground at 0.93c. (a) What is the mean lifetime of a muon in the reference frame of the ground? (b) What is the distance to the ground from the perspective of the muons? (c) What fraction of the muons will survive to reach the ground? (d) What fraction would reach the ground if classical mechanics were valid? 6. A pion is an elementary particle that, on average, disintegrates 2.610-8 s after its cre- ation in a frame at rest relative to the pion. An experimenter finds that pions created in a laboratory travel 13 m on average before disintegrating. How fast are the pions trav- eling through the lab?