Relativity With velocity of objects moving in different fram

[David0709]

Homework Statement

1. i)A police spaceship P is chasing another spaceship A. Both ships have velocities βP = βA = 3c/5 as measured along the x-axis in the Solar System reference frame O. The police ship is a distance L = 1 light-second (i.e. the distance traveled by light in one second) behind ship A. What is this distance in the frame of the police ship O′?
   
   ii)The police ship fires a missile at speed βM′ = 4c/5 at time t′ = 0 and position x′ = 0 as measured in the police spaceship frame O′. When and where does the missile reach ship A, as measured in frame O?
   
   Really I am interested to see If my solution is correct and if not where i went wrong.

Homework Equations

t′ =γ(ct−βx), x′ =γ(x−βct), y′ =y, z′ =z And all inverses as well

D = s * t
γ = 5/4 (if you calculate it)

The Attempt at a Solution

i) I believe that length contraction is a valid way to proceed and in the frame of the solar system since it is moving relative to observer it will be not proper length while in frame O' it is proper length hence in O' it should be 5/4 light second

ii) Since moving at same speed and using addictive relativistic formula the relative velocity is zero between the two objects and in frame O' they are separated at 5C/4 and the rocket being fired relative to police spaceship is 4c/5 hence the time taken in frame O' , t' = 25/16 and x' = 5c/4

Using the inverse lorentz equation (to convert position in frame O' to O) we see that x = 175c/64 and t = 185/64

Please could anyone let me know if the above is correct?
Thanks

 

[vela]

Looks good except for the lack of units on your answers.

You could check your answers by solving the problem in the unprimed frame by calculating the speed of the missile in the unprimed frame and then solving for when it'll catch up to the moving ship A.