An alien spaceship traveling at 0.630 c toward the Earth launches a landing craf

Question

An alien spaceship traveling at 0.630c toward the Earth launches a landing craft. The landing craft travels in the same direction with a speed of 0.790c relative to the mother ship (as measured by aliens on the mother ship). As measured on the Earth, the spaceship is 0.100 ly from the Earth when the landing craft is launched.     (a) What speed do the Earth-based observers measure for the approaching landing craft?
     c
    
    (b) What is the distance to the Earth at the moment of the landing craft's launch as measured by the aliens?
          ly
    
    (c) What travel time is required for the landing craft to reach the Earth as measured by the aliens on the mother ship?
          yr
    
    (d*) What travel time is required for the landing craft to reach the Earth as measured by Earth-based observers?
          yr
    
    (e*) What travel time is required for the landing craft to reach the Earth as measured by those on the landing craft?
          yrxc

Explanation / Answer

Part A)

u' = .79c

u = ?

v = .63c

Apply u = v+u'/(1 + vu'/c^2)

u = .63c + .79c/(1 + .63(.79)

u = .948c

Part B)

Apply L = L' (sqrt (1-v^2/c^2)

L = (.1)(sqrt (1 - .63^2)

L = .0777 ly

Part C)

Apply t = t'/sqrt (1 - v^2/c^2)

t = .0336/(sqrt (1 - .79^2)

t = .0548 ly

Part D)
(.1) = (.948)(t)

t = .105 yrs

Part E)

For the craft,

L = (.1)(sqrt (1 - .948^2)

L = .03183 ly

Then d = vt

.03183 = (.948)(t)

t = .0336 yrs

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