An alien spaceship traveling at 0.630 toward the Earth launches a landing craft.

Question

An alien spaceship traveling at 0.630 toward the Earth launches a landing craft. The landing craft travels in the same direction with a speed of 0.700c relative to the mother ship (as measured by aliens on the mother ship). As measured on the Earth, the spaceship is 0.220 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 (mother ship)? ______ 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? _______ yr

Explanation / Answer

a) What speed do the Earth-based observers measure for the approaching landing craft? c
u' = (u - v)/(1 - uv/c2)
0.700 = (u - 0.630)/(1 - u*0.630)
u = 0.92297

u = 0.923 c

(b) What is the distance to the Earth at the moment of the landing craft's launch as measured by the aliens? ly
gamma = 1/(1-(v/c)^2)^0.5 = 1/(1-(0.630)^2)^0.5 = 1.28767
L = Lp/gamma = 0.220/1.28767 = 0.17085

L = 0.171 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
t = 0.17085/0.630

t = 0.2712 yr

(d*) What travel time is required for the landing craft to reach the Earth as measured by Earth-based observers? yr
t = 0.220/ 0.92297 = 0.2383 yr

(e*) What travel time is required for the landing craft to reach the Earth as measured by those on the landing craft? yr
t = gamma * t_p
t_p = t/gamma = t * (1-(u/c)^2)^0.5

= 0.238*(1- 0.92297*0.92297)^0.5

= 0.0915 yr

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