Is There Enough Information to Solve the Relativistic Velocity Addition Problem?

[mfreeman]

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]

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

Relativistic addition of velocities:
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".)

 

[mfreeman]

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]

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]

Thus we have 0.50c = 2V' / (1 + (V'^2/c^2)).

Actually, this equation looks fine to me.

 

[mfreeman]

Gosh I thought so! Thanks.

 

[Doc Al]

Gosh I thought so! Thanks.

Good!

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

 

[Ray Vickson]

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.