If relativistic train acclerates, what is the speed of the last car?

Question

Locomotive pulls 500,000 cars, each car 1km long.

In 1 second locomotive accelerates from 0km/s to
4/5 c = 240,000 km/s, total distance moved by the
locomotive is 120,000 km.

At this speed each car shrinks to √(1 - (4/5)²) = 3/5 km.

What distance did the last car travel in 1 second?

Answer 1

I have a feeling that this is a trick question, and that you really know the answer. :-)

At first glance, it would seem that the last car has moved 320,000 km in one second (which of course is faster than the speed of light(!)) At t=0, the first car is at km marker 0, and the last car is at km marker "-500000". at t=1, the first car is at km marker 120000, and the last car (supposedly) is at km marker "-180000".

But the real answer is that the last car has not moved at all during the first second. That's because the force exerted by the locomotive cannot propagate down the train at a rate faster than c (in fact, it propagates no faster than the speed of sound through whatever material the cars are made of). So essentially, the train stretches during the first second (assuming it is strong enough to keep from breaking!)

Answer 2

The speed of propagation of an impulse from the locomotive to the last car is going to be at something on the order of the speed of sound, not the speed of light. Assuming perfect stiffness, speed of impulse = c, the caboose would be waiting for over a second. In reality, it is going to be waiting for quite a while longer than that.

Answer 3

I don't think that in one second, the last car has even got the memo yet that the locomotive is moving. Even if the signal to get going traveled at light speed, it takes 5/3 seconds to get back there.

So the answer is zero.

Answer 4

Who measures this 1 second?
Somebody in the last car or somebody in the locomotive?
The answer will be different depending on where the person who measures is, hence the term "relativistic".