Speed of light, c = 3 × 10 8 m/s.
Lorentz Transformations: x
0
= γ ( x x = γ ( x
0
− vt
+ vt
0
)
) t t
0
= γ ( t −
= γ ( t
0
+ vx c 2 vx
0 c 2
)
)
γ =
1 q
1 − v
2 c
2
Relative Velocity: u
0
=
1 u − v
− uv c
2
Law of Addition of Velocities: u = u
0
1+
+ v u c
0
2 v
Doppler effect (for light from a receding source): f = f
0 q
1 −
1+ v c v c
Please write your answers in your blue book, and make sure your secret code number is written on all pages in indelible ink.
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1. Consider two systems of reference S and S
0 such that, relative to S , S
0 is in uniform motion with velocity v . Two events E
1 systems. In S
0 the events occur at the same place and E
2 are observed in both but are separated by a time interval of 4 s. In S they occur at different places and are separated by a time interval of 6 s.
(a) What is the value of v ?
(b) What is the distance by which the two events appear separated in S ?
2. At noon a rocketship pass the Earth with a velocity of 0 .
8 c . Observers on the ship and on the Earth agree that it is noon.
(a) At 12:30 p.m., as read by the rocketship clock, the ship passes an interplanetary station that is fixed relative to the Earth and whose clocks read the Earth time. What time is it at the station when the ship passes by?
(b) How far from the Earth (in Earth coordinates) is the station?
(c) At 12:30 p.m. rocketship time, the ship reports by radio back to Earth.
When (by Earth time) does the Earth receive the signal?
3. A K 0 meson at rest decayed into a π + meson and a π − meson, one flying east at a speed of 0 .
6 c and the other flying west at the same speed of 0 .
6 c . This phenomenon was also watched by an observer S ; the K 0 meson was moving east at a speed of 0 .
8 c during its decay with respect to the observer. What were the velocities of the π mesons observed by S ?
4. There is a spaceship shuttle service from Earth to Mars. Each spaceship is equipped with two identical lights, one at the front and one at the rear. All spaceships travel at the speed v relative to the Earth, with the result that the headlight of a spaceship approaching the Earth appears green (with λ = 500 nm) while the taillight of a spaceship leaving the Earth appears as red (with
λ = 600 nm).
(a) What is the value of v ?
(b) What is the proper wavelength of the lights?
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