Two rockets are each 1000m long in their rest frame. Rocket 1,traveling at .9*c

Question

Two rockets are each 1000m long in their rest frame. Rocket 1,traveling at .9*c relative to earth, in over taking rocket 2.Rocket 2 is only moving at .5*c. According to the crew on rocket 2,how long does it take for rocket 2 to completely pass?
So far I have calculated the speed of rocket 1 in refferance torocket 2, to be 2.18181818*10^8m/s. I am just not sure on thelength contraction to find out the amount of time it takes forrocket 1 to completely pass in refferance to ship 1.
Masteringphysics problem.

Explanation / Answer

You need to figure out the relative velocity between Ship 1and Ship 2 See Wiki: Special Relativity composition velocities . Using Wiki terminology: Rocket 1 speed: w = .9c Rocket 2 speed: v =.5c Speed differential in reference frame of Rocket 1
w' = (w-v)/(1 -wv/c^2) w' = .7273 c . Length contraction of R1 in R2's frame L.r1 = L * (1- (.7273)^2) T = (L + L.rl)/w' Note: (L + L.rl) is the distance from when the tip of R1reaches the tail of R2 until the tail of R1 reaches the tip ofR2 T = 7.73x10^-6 s

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