(a) A space vehicle travels at 100,000 m/s relative to the earth. How much time will its clocks gain or lose, as compared to earth-based clocks, in a day? ( points) (b) What must be one’s speed, relative to a frame S , in order that one’s clocks will lose 1 second per day as observed from S? What if they are to lose 1 minute per day? (9 points)
Muons are subatomic particles that are produced several miles above the earth’s surface as a result of collisions of cosmic rays with atoms in the atmosphere. These muons rain down more-or-less uniformly on the ground, although some of them decay on the way since the muon is unstable with a proper half-life of about 1.5 s. In a certain experiment a muon detector is carried in a balloon to an altitude of 2000 m, and in the course of 1 hour it registers 650 muons traveling at 0.99 c toward the earth. If an identical detector remains at sea level, how many muons would you expect it to register in 1 hour? This was essentially the method used in the first tests of time dilation, starting in the 1940’s. (10 points)
A rigid spherical ball (rest frame S ) is observed from a frame S’ that travels with speed 0.5 c relative to frame S. Describe the ball’s shape as measured by observers in S’. (8 points)
In a frame S , two events have spatial separation Δ x = 600 m, Δ y = Δ z = 0, and temporal separation Δ t = 1 s. A second frame S’ is moving along Ox with nonzero speed v and O’x’ parallel to Ox. In S’ it is found that the spatial separation Δ x’ is also 600 m. What are v and Δ t’? (10 points)
Suppose that as seen in a frame S , a signal (a pulse of light, for example) has velocity c along the y axis. (14 points) (a) Write down the components of its velocity u’ relative to a frame S’ traveling in the standard configuration with speed v along the x axis of frame S. (6 points) (b) In what direction is the signal traveling relative to S’? (4 points) (c) Using your answer to part (a), calculate the magnitude of u’. Comment on the result. (4 points)

As seen from earth (rest frame S ) two rockets A and B are approaching in opposite directions, each with speed 0.9 c relative to S. Find the velocity of rocket B as measured by the pilot of rocket A. (10 points)
At the Stanford Linear Accelerator, electrons are accelerated to energies of 50 GeV. (9 points) (a) If this energy were classical kinetic energy, what would be the electrons’ speed? (3 points) (b) Calculate γ and hence find the electrons’ actual speed. (6 points)
A lambda particle (Λ ) decays into a proton and a pion, Λ  p + , and it is observed that the proton is left at rest. The masses involved are m Λ = 1116, m p = 938, and m  = 140, all in MeV/ c^2. (12 points) (a) What is the energy of the pion? (8 points) (b) What was the energy of the original Λ? (4 points)
A neutral pion traveling along the x axis decays into two photons, one being ejected exactly forward, the other exactly backward. The first photon has three times the energy of the second. Find the speed of the original pion. ( points)