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My question is based on a popular question, usually posed in introductory courses on the special theory of relativity.

Two rockets are on a collision course. Rocket 1 is traveling at the speed 0.8c while Rocket 2 moves at the speed 0.6c, both relative to the Earth. The rockets are $2.52 \times 10^{12}m $ apart as measured by Lucy at rest on Earth. Both rockets are 50 m as measured by Lucy on Earth.

The question usually has many parts regarding the lengths of the rocket, speed of the rockets from the frame of the other etc. My issue is with the question regarding the time taken for the rockets to collide in each frame. There are two methods I can think of .

a.First measure the time taken for collision in Lucy's frame (comes up to be 100 minutes ) and then account for time dilation and calculating the proper time (i.e. divide by $\gamma$ ).

b.The other approach is to calculate the distance in the frame of one of the rockets and then calculate the time for collision by using the speed of the other rocket in the frame of the first rocket.

The issue is that the two approaches lead to different answers. So, my question is which method is correct and please explain the why one approach works and the other does not.

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  • $\begingroup$ "Time taken for collision in Lucy's frame" between what event and the collision? You always need a pair of events to measure the time between, and if you choose an event that isn't on Lucy's worldline, then you have to take into account the relativity of simultaneity (from what I've seen, when someone is confused about a basic problem in SR, 99% of the time their confusion has to do with failing to take into account the relativity of simultaneity). $\endgroup$ – Hypnosifl Nov 16 '14 at 16:58
  • $\begingroup$ I considered the issue of simultaneity. But I couldn't really arrive at the answer. The only start point we have is the time at which the distance in Lucy's frame is the number i mentioned. $\endgroup$ – Abhishek Shetty Nov 16 '14 at 17:01
  • $\begingroup$ In Lucy's frame you can write down coordinates of two events, one on each rocket's worldline, which are simultaneous in her frame, and which have the specified spatial separation. Then in her frame, the time between either event and the collision is 100 minutes. For rocket #1, you can use the time dilation formula to find the time between the event you chose on rocket #1's worldline and the collision, since both these events could be measured by a clock at rest in rocket 1's frame, but you can't use it for the time between the event on rocket #2's worldline and the collision in #1's frame. $\endgroup$ – Hypnosifl Nov 16 '14 at 17:11
  • $\begingroup$ And again, "the time taken for the rockets to collide" is a meaningless question unless you specify what events you want the time between in whatever frame you're using. If this is a problem you've seen in an introductory course or text, can you quote the actual wording? $\endgroup$ – Hypnosifl Nov 16 '14 at 17:12
  • $\begingroup$ The exact phrasing is : "According to Rocket 1, how long will it take to collide?" I found this on a course assignment sheet for which I can provide the link. The question I had in my course had more fluff but in essence it was the same. link $\endgroup$ – Abhishek Shetty Nov 16 '14 at 17:17
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So, my question is which method is correct and please explain the why one approach works and the other does not.

For method one, Liz knows the initial separation and individual rocket velocities so the calculation of the time to impact is straightforward.

The most likely problem with method two is due to the relativity of simultaneity. This is a subtle point that will be clear if you draw a spacetime diagram.

One cannot apply the length contraction formula to the initial separation in Liz's frame. The context in which the length contraction result is derived assumes the length $L$ is constant in time.

However, the separation distance changes with time and thus, due to the relativity of simultaneity, this must be accounted for.

enter image description here

If $d_0$ is the initial distance between the rockets according to Liz, then the initial distance according to Rocket 1 is

$$d'_0 = \frac{d_0 \sqrt{1 - (0.8)^2}}{1 + (0.8)(0.6)}$$

while the speed of Rocket 2 according to Rocket 1 is

$$v'_2 = \frac{0.6 + 0.8}{1 + (0.8)(0.6)}c$$

such that the time to collision, according to Rocket 1 is

$$\Delta t' = \frac{d'_0}{v'_2} = \frac{d_0}{(0.6 + 0.8)c}\sqrt{1 - (0.8)^2} = \Delta t \sqrt{1 - (0.8)^2} $$

where $\Delta t$ is the time to collision in Liz's frame.

But this is precisely the result from method one. So, the two methods do in fact agree when method two is done properly.

To summarize, naively applying length contraction to the initial separation distance in Liz's frame will give an answer different from method 1. The reason for this is that the separation distance is not constant in any frame. Thus, the length contraction formula cannot be applied.

As I mentioned earlier, a spacetime diagram will make this clear.

enter image description here

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