PHYS 3310 Module 1 Self Assessment Practice Problems

Module 1 Self Assessment Practice Problems

1.1
If relativistic effects are to be less than 1%, then 𝛾 must be less than 1.01. At what relative velocity is 𝛾 = 1.01?
Answer: 0.14c
1.2
Inside a spaceship flying past the earth at three-fourths the speed of light, a pendulum is swinging.
(a) If each swing takes 2.00 s as measured by an astronaut performing an experiment inside the spaceship, how long will the swing take as measured by a person at mission control (on earth) who is watching the experiment?
(b) If each swing takes 2.00 s as measured by a person at mission control, how long will it take as measured by the astronaut in the spaceship?
Answer: (a) 3.02 s (b) 1.32 s
1.3
Two atomic clocks are carefully synchronized. One remains in New York, and the other is loaded on an airliner that travels at an average speed of 230 m/s and then returns to New York. When the plane returns, the elapsed time on the clock that stayed behind is 3.50 h.
(a) By how much will the readings of the two clocks differ? (Hint: Since u << c, you can simplify by a binomial expansion.)
(b) Which clock will show the shorter elapsed time?
Answer: (a) 3.7 ns (b) the moving clock
1.4
The proper lifetime of a certain particle is 120.0 ns.
(a) How long does it live in the laboratory if it moves at v = 0.950c?
(b) How far does it travel in the laboratory during that time?
(c) What is the distance traveled in the laboratory according to an observer moving with the particle?
Answer: (a) 384 ns (b) 109 m (c) 34.2 m
1.5
A space probe is sent to the vicinity of the star Capella, which is 42.2 light years from the earth. (A light year is the distance light travels in a year.) The probe travels with a speed of 0.9930c. An astronaut recruit on board is 19 years old when the probe leaves the earth. What is her biological age when the probe reaches Capella?
Answer: 24 years old
1.6
A cube of metal with sides of length a sits at rest in a frame S with one edge parallel to the x-axis. Therefore, in S the cube has volume a3. Frame S’ moves along the x-axis with a speed u. As measured by an observer in frame S’, what is the volume of the metal cube?
Answer: a3(1-u2/c2)1/2
1.7
How fast must an object move before its length appears to be contracted to one-half its proper length?
Answer: 0.866c
1.8
Space pilot Mavis zips past Stanley at a constant speed relative to him of 0.800c. Mavis and Stanley start timers at zero when the front of Mavis’s ship is directly above Stanley. When Mavis reads 5.00 s on her timer, she turns on a bright light under the front of her spaceship.
(a) Use the Lorentz coordinate transformation to calculate x and t as measured by Stanley for the event of turning on the light.
(b) Use the time dilation formula to calculate the time interval between the two events (the front of the spaceship passing overhead and turning on the light) as measured by Stanley. Compare to the value of t you calculated in part (a).
(c) Multiply the time interval by Mavis’s speed, both as measured by Stanley. to calculate the distance she has traveled, as measured by him when the light turns on. Compare to the value of x you calculated in part (a).
Answer: (a) 2 x 109 m (b) 8.35 s
1.9
A pursuit spacecraft from the planet Tatooine is attempting to catch up with a Trade Federation cruiser. As measured by an observer on Tatooine, the cruiser is traveling away from the planet with a speed of 0.600c. The pursuit ship is traveling at a speed of 0.800c relative to Tatooine, in the same direction as the cruiser.
(a) For the pursuit ship to catch the cruiser, should the velocity of the cruiser relative to the pursuit ship be directed toward or away from the pursuit ship?
(b) What is the speed of the cruiser relative to the pursuit ship?
Answer: (a) toward (b) 0.38c
1.10
Two spaceships approach the Earth from opposite directions. According to an observer on the Earth, ship A is moving at a speed of 0.743c and ship B at a speed of 0.831c. What is the velocity of ship A as observed from ship B? Of ship B as observed from ship A?
Answer: 0.973c