# Relativistic velocity problem with x and y components

1. Sep 30, 2015

### Luke Cohen

1. The problem statement, all variables and given/known data
Two rockets leave their space station along perpendicular paths, as measured by a flight controller on the space station (see figure). The flight controller records the speeds of Rockets 1 and 2 to be 0.6 c and 0.8 c respectively. What is the velocity of Rocket 2 as measured in the reference frame of Rocket 1? Give both the components and the magnitude of this velocity

Rocket one is going in the positive y direction and rocket 2 is going in the positive x direction.

2. Relevant equations

3. The attempt at a solution

So I tried setting it up by using rocket 1 as the S' frame and rocket 2 as the S frame. The components of velocity are: S' = (-0.8c, 0.6c, 0) and S = (0.8c, -0.6c, 0). Plugging these values into the Ux equation, -0.8 + -0.8 / (1+0.64), but the correct answer is 0.64c for Ux. I also need to figure out Uy, but I think if you can help me solve for Ux, then I can solve for Uy myself. I appreciate all help, Thanks!

2. Sep 30, 2015

### Ray Vickson

You can write the Lorentz transformation T1 for (x,y,t) to (x1,y1,t1) with velocity v1 along the x-axis, and the Lorentz transformation T2 for (x,y,t) to (x2,y2,t2) with velocity v2 along the y-axis. To get the transformation for (x1,y1,t1) to (x2,y2,t2), just express express (x2,y2,t2) in terms of (x,y,t) and then express (x,y,t) in terms of (x1,y1,t1). From that, you can work out the relative velocity. However, it will be messy, so get out several sheets of paper and a sharp pencil. Alternatively, you can use a computer algebra system to make it manageable.

3. Sep 30, 2015

### Luke Cohen

I don't think my professor would assign anything requiring a computer algebra system.... I am also sure that I should be able to solve this problem with the LT equation above for Ux and then the LT equation for Uy. Is there something I am doing incorrectly with my assigning of values to the variables U'x or V?

4. Sep 30, 2015

### Ray Vickson

If (x1,y1,t1) is obtained from (x,y,t) by a Lorentz transformation T1 with velocity v1 along the x-axis, then coordinates of particle 2 (relative to particle 1) are obtained by putting (x,y,t) = (0,v2t,t) in the transformation equation T1. From that you can get the velocities in the 1-frame.

Last edited: Sep 30, 2015
5. Oct 1, 2015

### vela

Staff Emeritus
I don't see how you got those. It's not even clear what frame you're saying these velocities would be observed in. Note that you're also claiming that S and S' are both moving with speed $c$ relative to whatever rest frame you're using since (0.8)^2+(0.6)^2 = 1.