A relativistic train of rest length 960 meters travels at 0

Physics 252 Homework
Due Friday April 2 1999
Let’s have another look at the midterm!

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?

That was pretty straightforward, right?

(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?

Now here you should have had a mental image of the train frame as a set of axes firmly attached to the train, with the origin at the front of the train, since it says above that (0,0) in both frames marks the front of the train entering the tunnel. If the front of the train is at the origin, the back of the train isn’t. it’s evidently at some negative value of x.

(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.

NOW DO THIS FOR THE CURRENT HOMEWORK ASSIGNMENT:

1. Imagine there’s a parallel railroad track some distance away that does not go through the tunnel, but instead goes by the side of the mountain. Now suppose you are on a train, call it train 2, on this parallel track that is travelling at 0.4c.

(a) What is the length of the tunnel as viewed from this frame?

(b) What is the length of train 1 as viewed from train 2? (Be careful! This is a trick question! The answer is not the same as (a).)

Back to the midterm …

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.

Just about everybody bombed on this one. I was a little upset, because I went through it in class, in talking about particle creation. But I now realize you didn’t work any examples yourselves, so, we’ll remedy that!

DO French Chapter 7, problems 2 and 3.

Most of the rest of the midterm was better done, except that very few of you realized that the cooling effect on a gas in an expanding container is explained in the kinetic theory by the molecules losing energy as they bounce off a receding wall. This means you didn’t get the point of our cool applet illustrating this! The kinetic theory is about motion, that’s what kinetic means. Just stating the gas laws doesn’t give any clue why the gas cools, and is completely useless in understanding why a fast receding wall might not cool the gas.

READ Evolution of the Atomic Concept and the Beginnings of Modern Chemistry and answer the following:

4. (a) Why did Avogadro come up with his hypothesis?

(b) Consider two identical containers, one filled with helium, one with argon at the same temperature and pressure. Show that the average kinetic energy of the helium atoms will be the same as that of the argon atoms if and only if Avogadro’s hypothesis is correct.

5. Why does the idea that matter is composed of atoms imply that electrical charge is quantized, that is, it only comes in multiples of a certain value?

6. At the end of the lecture, there’s a quote from Ostwald:

"The proposition that all natural phenomena can ultimately be reduced to mechanical ones cannot even be taken as a useful working hypothesis: it is simply a mistake. This mistake is most clearly revealed by the following fact. All the equations of mechanics have the property that they admit of sign inversion in the temporal quantities. That is to say, mechanical processes can develop equally well forwards or backwards in time. Thus, in a purely mechanical world there could not be a before and an after as we have in our world: the tree could become a shoot and a seed again … . The actual irreversibility of

natural phenomena thus proves the existence of processes that cannot be described by mechanical equations …"

Restate this idea briefly in your own words, and comment on its reasonableness and its validity.