2. Relativistic snowballs. (40 points) You are at the origin (x, y) = (0, 0) in your frame S. Your friend, moving along the positive x-axis at velocity , throws a snowball of mass m at you at the instant he is located at (x, y) = (-L, -d) in your frame. In your friend's frame S', the snowball is thrown with kinetic energy 3m. (a) At what angle with respect to the x-axis in S' must your friend aim the snowball in order to hit you? (b) How much time elapses in your frame before the snowball hits you? (c) How much proper time elapses in the snowball's frame before the snowball hits you? (d) What is the magnitude of the snowball's momentum, in both frames S' and S? 3. Relativistic stoplight. (20 points) (a) You are in a spaceship on an intergalactic superhighway and you see a traffic light up ahead. You observe the light to be green (wavelength 550 nm) so you continue on your way. But you are pulled over by the space police, who tell you that you ran a red light! If the stoplight did indeed emit red light (wavelength 700 nm) in its rest frame, how fast were you going? (b) To show that this situation never arises in everyday life, calculate the smallest wavelength a human could have observed from a 700 nm red light, traveling relative to the light at the fastest speed humans have ever achieved, 40,000 km/hr (measured during the re- entry of Apollo 10 in 1969). Before you start, compare this speed to the speed of light, and consider whether a Taylor expansion of the Doppler formula would be appropriate!