Two spaceships, each 100 m long when measured at rest, travel toward each other

Question

Two spaceships, each 100 m long when measured at rest, travel toward each other with speeds of 0.85c relative to Earth. (a) How long is each ship as measured  by someone on Earth? (b) How fast is each ship traveling as measured by an observer on the other? (c) How long is one ship when measured by an observer on the other? (d) At time t=0 on Earth, the fronts of the ships are together as they just begin to pass each other. At what time on Earth are their ends together? (e) Sketch accurately scaled diagrams in the frame of one of the ships showing the passing of the other ship.

Explanation / Answer

(a) speed of each spaceship relative to the earth = 0.85c = v= w    Now, = 1/[1 - (v/c)2] =1.89831     So, apparent length = L = Lo/ =100/1.89831 m = 52.6783 m (b) relative speed = r = [v - (-w)] / [ 1 -(v*(-w))/c2] = [ 1.7c]/[1 + 0.7225] = 0.9869376 c (c) Apparent length of one ship from the other     L' = Lo / '    ' = 1/[1 -(r/c)2] = 6.20721    => L' = 100 / 6.20721 m = 16.1103 m (d) time taken on the ships = L'/r = 16.1103 /0.9869376*3*108 s = 5.4412*10-8 s     And, time taken on earth = L/v = L/w =52.6783/0.85*3*108 s = 2.06582*10-9s

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