In trying to come up with an example in S.R., I came up with the following: 2 rockets, a distance d apart as measured in the earth frame, each have a speed of 0.8c and are on a head on collision with the other. What is the time to impact as measured in the earth frame and on one of the rockets. As first this did not seem like a difficult example, but when I tried to change the speed of one of the rockets to 0.4c, that's when the conflicts in my mind began. Here is how my original solution follows and then why it bothers me:(adsbygoogle = window.adsbygoogle || []).push({});

In the earth frame, the point of collision is at the origin IF you start each rocket d/2 on either side. Thus, an observer on earth would measure the time to impact as t=(d/2)/0.8c, the time it takes either rocket to travel a distance of d/2 at a speed 0.8c. This yields t=0.625(d/c). To find the collision time in the rocket frame, it seems to me that I have 3 possible methods:

1) simple time dilation

2) lorentz transformations

3) t=d/v in the rocket frame

1) Using method 1, since the rocket is measuring proper time, I use t=(1/gamma)*t(in earth frame), with gamma calculated with v=0.8c. This yields a time to collision of 0.375(d/c).

2) Using method 2, with the event coordinates as:

S (0,0,0,0) initial

S (d/2,0,0,0.625(d/c)) final

S' (0,0,0,0) initial

S' (0,0,0,t') final

the t' L.T. equation yields 0.375(d/c) - same as method 1

3) In finding t=d/v in the rocket frame, I used the velocity transformation equation for Vx and got -(40/41)c. Using the rocket on the left (at -d/2 earth frame), the left rocket sees the distance between the rockets as being contracted by gamma*d. Again using v=0.8c in gamma, I get a contracted length of 0.6d. Calculating d/v yields 0.615(d/c), which is not the same as before. If I use (40/41)c for gamma which I think is correct, the result is 0.225(d/c) - still not equal.

It seems to me that all these methods should yield the same collision time for either rocket. I have not had S.R. in almost 15 years, so I am really rusty on this and hope someone more gifted in this area can see my flaw and/or mis-understanding.

This example was modified from a homework problem I came across in which a fixed time to evacuate a ship is 1.5 hrs. By calculating the collision time in each rocket frame (one at 0.8c the other at -0.6c), is there enough time to get every one off each rocket before the collision? The book had an earth collision time of 1.67hrs, rocket A at 1.47 hrs and rocket B at 1.97 hrs. Thus, the crew of rocket A don't make it.

However, my problem with this approach is that I would think that both rockets clocks should have collisions times slower than that of the earths? If you view this result in terms of tick rates of the clocks (R=1/t), rocket A would have a faster tick rate than the earth clock - which goes against the relativity statement that "moving clocks tick slower". It was at this point that the meltdown occurred. If anyone has a clue to the resolution of my paradox, I would greatly appreciate it. Thanks in advance.

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# A paradox of my own

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