# Velocity Transforms

Skullmonkee

## Homework Statement

A spaceship moves away from a point at speed v and launches a smaller ship at speed v relative to the first ship. The second ship shoots a missile at speed v relative to the second ship
1. what is the speed of the second ship relative to Earth
2. what is the speed of the missile relative to Earth

## The Attempt at a Solution

I assume that in order to get the speed of the first ship relative to Earth i just use the velocity transform u=(u'+v)/(1+[vu'/c^2]), which gives u= (v+v)/(1+[v^v/c^2]) or u= 2v/(1+[v^2/c^2])

Is this correct?
If so, to find the speed of the missile relative to Earth would i again use the same equation using the 2v/(1+[v^2/c^2]) as u'? Would this give the answer to part 2?
Im confused.

Last edited:

Homework Helper
I think that's what you are supposed to do, yes.

fredrick08
hi can i ask, since i am doing same question.. i understand part a.. just simple transforms, but what did u mean by using u=2v/(1+v^2/c^2) for b?? then the answer for b would be... solving for u'?

Skullmonkee
Well my thinking was that because the speeds are not a number value (they are given as 'v'), to find the answer for part 1 you must find u and simplify. This gives u= 2v/(1+[v^2/c^2]).
For the second part the u that was found in 1. becomes u' in part 2. (Sorry i forgot to add the ' to the u)

Now I am just confusing myself and I am not sure if it is right or not.

fredrick08
ya that's wat i was thinking... btu when u do that calculation, u get a weird answer... but anyways thanks for ur help

Homework Helper
Since you both seem to be on the right track, the answer I get (for you to compare to) is

$$\frac{ 3 v + (v^2 / c^2) v }{ 1 + 3 (v^2 / c^2) }$$

Yes, it looks a little weird.

Writing it in terms of $\beta = v / c$ gives the still a bit weirdly looking

$$\frac{ \beta(3 + \beta^2) }{ 1 + 3 \beta^2 } c$$

Skullmonkee
Thanks CompuChip. but how did you derive that answer? I tried doing it myself but got really lost.

Homework Helper
I just plugged u = 2v/(1+v^2/c^2) back into the addition formula and simplified it (ok, maybe I cheated a bit by letting the computer simplify it).

Where did you get stuck?

Skullmonkee
I got stuck in the algebra. After plugging it into the formula as you did.

fredrick08
sorry didnt see the c... = (