1. A relativistic train of rest length 960 meters travels at 0.8c through a tunnel which also has rest length 960 meters. The train enters the tunnel through door 1, and leaves by door 2.

The frame of reference of the tunnel we label S_tunnel, and denote events (x, t).

The frame of reference of the train we label S_train, and denote events in that frame (x¢ , t¢ ).

We label the event of the front of the train passing door 1 as event F1, and choose our axes by making that event (0,0) in both frames.

For convenience, we take as our unit of time the microsecond, 10^-6 seconds, so the speed of light is 300 meters per microsecond.

(a) If F2 labels the event of the front of the train passing door 2, what are the coordinates (x, t) of F2 in S_tunnel?

 (b) If B1 labels the event of the back of the train entering the tunnel, what are the coordinates in the train frame of reference (x¢ , t¢ ) of B1?

 (c) Write down the Lorentz equations, and use them to find the coordinates of B1 in S_tunnel. Is B1 before or after F2?

 (d) Use the Lorentz equations to find the coordinates of F2 in S_train. Is B1 before or after F2?

 (e) Is there a frame of reference in which the events B1, F2 are simultaneous? If your answer is no, give a reason. If it’s yes, find the frame, and describe briefly how the train and tunnel look in that frame.

 2. Suppose an accelerator can give a proton a kinetic energy of 1 Tev, that is, 1000Gev. Taking the rest mass of a proton in energy units to be 1 Gev (that is, 10⁹ ev), find the maximum possible rest mass of an unknown particle X that could be produced in a collision

p + p ® p + p + X

where the second p on the left hand side of this equation is in a fixed target. Since the value used for the proton mass is a few percent off, you can make approximations of that same order, but state any such approximations you make.

 How would your answer change if the two p’s were in colliding beams, each having kinetic energy 1 Tev, colliding head on?

3. (a) Give a brief statement of the law of equipartition of energy.

(b) Estimate the total amount of heat energy in one mole of helium gas at room temperature.

  (c) Given that the atomic weight of helium is close to 4, estimate the average speed of helium atoms in the gas at room temperature.

  (d) Argon is a monatomic gas like neon, but with an atomic weight of 40. Answer (b) and (c) above for Argon.

  (e) Suppose a gas is in a piston type container, and the piston is slowly moved to give the gas more room. Explain why the kinetic theory of gases predicts that the gas should cool.

  (f) If the piston is moved fast enough, the cooling effect becomes small. How fast is fast enough? Explain, using the kinetic theory.

4. (a) Sunlight at the earth’s surface supplies energy at a rate of approximately one kilowatt per square meter. Given that the distance from the earth to the sun is about 200 times the radius of the sun, what is the rate of energy radiation from one square meter of the sun’s surface?

  (b) Given that Stefan’s constant is 5.67x10^-8, estimate the temperature of the sun’s surface.

  (c) We have evolved to be most sensitive to electromagnetic radiation at the frequency where the sun’s rays are most intense. What, approximately, is that frequency?

  (d) From your answers to (b) and (c) above give an order of magnitude estimate of Planck’s constant. Full credit will be given for a correct argument leading to an answer within a factor of ten of the actual value.

5. (a) Write not more than five lines on how x-rays were discovered.

  (b) How was it established that x-rays were not the usual ultraviolet light?

  (c) Write a few lines describing how radioactivity was discovered. What was the discoverer looking for?

   (d) Why were alpha particles first thought to be uncharged?

  (e) What physical properties of cathode rays led to the belief that they were particles smaller than atoms, and that atoms might therefore be breakable?