A Question on Special Relativity

[dom139]

Homework Statement

You should take the velocity of light c = 3 x 105 km s^-1
Two spherical clusters of galaxies each have a diameter D = 10^20 km.
Cluster 1 is traveling at 258,000 kms^-1 relative to the Earth
while cluster 2 is traveling in the same direction at 171,000 kms^-1 .

Suppose that cluster 1 is just starting to pass through cluster 2 in the reference frame of the Earth.

How long would it take for cluster 1 to pass completely through cluster 2 in the reference frame of the Earth, assuming that there is no gravitational interaction between the two galaxy clusters.

Passing time = ? x D / c
Enter your answer to ONE decimal place as a multiple of D/c,
i.e. without entering values for D/c at this stage

Now evaluate your answer in millions of years.
= ? million years
Enter your answer to the nearest integer.

Homework Equations

The Attempt at a Solution

I've not done relativity since I was at college over a year ago, and now i have some coursework that is all on relativity and I have no idea what I'm doing, so any help or guidance would really be appreciated, thanks.

 

[voko]

What would be your approach to solve this if you ignored relativity?

 

[dom139]

I would work out how fast cluster 1 is moving relative to cluster 2, which is 87,000kms^-1.
And then i simply do time = distance/speed = 1.149E15.
So assuming I've gotten that right, i have to apply the relativistic factor somewhere?

 

[voko]

Basically your idea is to use cluster 2's reference frame. This gets complicated, because you are given (almost) everything in the Earth's frame, and the result is also required to be the Earth's frame. It is best to stay in the Earth frame.

Now, what you are not given in the Earth's frame is the diameters of the clusters, although this is not entirely clear from the description. I think you have to assume they are their proper diameters, so you have obtain their (Lorentz-contracted) diameters in the Earth's frame. Let's call them d1 and d2.

Let's say that the point where cluster 1 touches cluster 2 "face-to-back" at time t = 0 is x = 0. You need to find the time t = T such that at x = X cluster 1 touches cluster 2 "back-to-face". These x, X, t and T are all in the terrestrial frame. Can you find the relationship for X, T, d1, d2, v1 and v2?