Special Relativity, Spring '10

Homework 2

1. As in Mermin's example, suppose the ball moves east in the train frame at 5 ft/sec, and the train moves east in the track frame at 10 ft/sec. What is the ball's velocity in the track frame? Use equation (2.1) and be explicit about which thing is X, which is Y, and which is Z. (In other words, show me that you can derive the obvious and correct answer to this question using the equation.)

2. In the last few paragraphs of the chapter, Mermin raises the daunting possibility that maybe "one second of train time" and "one second of track time" aren't the same amount of time. Let's explore this further. Suppose it's just given that all the clocks on the train run slow by exactly a factor of two -- i.e., "one second of train time" is equal to "two seconds of track time". It will be convenient here to just define two distinct time units: the "train-second" and the "track-second". And then what we're just accepting as given is that they are related this way: 1 train-second = 2 track-seconds. Now finally the actual question: suppose (as in Mermin's example) that the ball moves east in the train frame at 5 ft/sec -- i.e., 5 ft/sec as measured by people on the train -- i.e., its speed is 5 ft/train-sec. And suppose that the train moves east in the track frame at 10 ft/sec -- i.e., its speed is 10 ft/track-sec. What is the speed of the ball in the track frame?

3. In this same weird scenario (where 1 train-second = 2 track-seconds), what is the speed of the track in the train frame? Does this contradict Mermin's equation (2.2) and the discussion surrounding it?

4. Same sort of deal, but let's change the numbers. Suppose the train is moving at 2 ft/sec in the track frame, and the ball is moving at 5 ft/sec in the train frame. But now both the time and distance units will be distorted. As before, let's say 1 train-second = 2 track-seconds. And similarly, suppose that 5 train-feet = 6 track-feet. Question: what is the speed of the ball in the track frame? Why is this interesting?

5. When the time and distance units in the two frames are different (as in the last couple of questions), is Equation (2.1) still true? Explain why or why not.

6. We've been contemplating the possibility that (e.g) "one second of track time" and "one second of train time" might not be equal. But... come on! What in the world could that even mean? For example, suppose there's a grandfather clock on the train, and you look at it through the window as it passes you. (You're standing at the station.) What would it look like? What would goings-on on the station platform look like if you were on the train looking out the window?