## Homework Statement

Two asteroids are approaching one another moving with the same speed speed as measured from a stationary observer on Europa. Their relative speed is 0.5c. Find the speed of one of asteroids relative to Europa.

I understand how relativistic velocity addition works but am not able to solve this question. Is it possible there is not enough information given in the problem? Any help would be much appreciated.

## Homework Equations

u = (u' + v) / (1 + (u'v)/c^2)

## The Attempt at a Solution

The ambiguity of the problem statement has led me down several different paths each resulting in a quadratic that in most cases results in a speed faster than c.

Doc Al
Mentor
You might find the following form of the equation easier to understand.

$$V_{a/c} = \frac{V_{a/b} + V_{b/c}}{1 + (V_{a/b} V_{b/c})/c^2}$$

Hint: Let "b" be Europa. ("a/c" means the velocity of "a" as measured in the frame of "c".)

Assuming "b" is Europa I would let V(a/c) be the speed of Ship A relative to Ship B which we know to be 0.50c. By the problem statement we have V(a/b) = V(b/c) = V(c/b) = V'. Thus we have 0.50c = 2V' / (1 + (V'^2/c^2)). This leads to a quadratic which I was told is the incorrect approach. Was that incorrect advice? Thank you very much for your help.

Doc Al
Mentor
Assuming "b" is Europa I would let V(a/c) be the speed of Ship A relative to Ship B which we know to be 0.50c.
To keep your sanity, let "a" stand for asteroid #1 and "c" stand for asteroid #2. Furthermore, let asteroid "a" move to the right and "c" move to the left. (As seen from Europa.)

By the problem statement we have V(a/b) = V(b/c) = V(c/b) = V'.
Careful! Signs matter. Let "to the right" be positive.

Doc Al
Mentor
Thus we have 0.50c = 2V' / (1 + (V'^2/c^2)).
Actually, this equation looks fine to me.

Gosh I thought so! Thanks.

Doc Al
Mentor
Gosh I thought so! Thanks.
Good!

Just for the record, if V(a/b) = V', then V(b/a) = - V'.

Ray Vickson
Homework Helper
Dearly Missed

## Homework Statement

Two asteroids are approaching one another moving with the same speed speed as measured from a stationary observer on Europa. Their relative speed is 0.5c. Find the speed of one of asteroids relative to Europa.

I understand how relativistic velocity addition works but am not able to solve this question. Is it possible there is not enough information given in the problem? Any help would be much appreciated.

## Homework Equations

u = (u' + v) / (1 + (u'v)/c^2)

## The Attempt at a Solution

The ambiguity of the problem statement has led me down several different paths each resulting in a quadratic that in most cases results in a speed faster than c.

There is no ambiguity. Relative to Europa, asteroid A moves at velocity +v and asteroid B at velocity -v. You are given that the velocity of B in the rest-frame of A is -c/2.