Question

please provide solutions for all questions; thank you in advance

An alien spaceship traveling at 0.610c toward the Earth launches a landing craft. The landing craft travels in the same direction with a speed of 0.840c relative to the mother ship (as measured by aliens on the mother ship). As measured on the Earth, the spaceship is0.260 ly from the Earth when the landing craft is launched.

(a) What speed do the Earth-based observers measure for the approaching landing craft?

(b) What is the distance to the Earth at the moment of the landing craft's launch as measured by the aliens (mother ship)?

(c) What travel time is required for the landing craft to reach the Earth as measured by the aliens on the mother ship?

(d*) What travel time is required for the landing craft to reach the Earth as measured by Earth-based observers?

(e*) What travel time is required for the landing craft to reach the Earth as measured by those on the landing craft?

Answer 1

here relativistics addition of velocity to be used.

part a:

speed of alien space ship w.r.t. earth=u=0.61 c

speed of landing craft w.r.t. alien space ship=v=0.84 c

then speed of landing craft w.r.t. earth=(u+v)/(1+(u*v/c^2))
=0.95874*c

part b:

distance observed by the aliens=original distance measured from earth

*sqrt(1-(v/c)^2)

where v=0.61*c

hence distance observed by alien ship=0.26*sqrt(1-0.61^2)=0.20602 light years

part c:

w.r.t. mother ship, speed of landing craft is 0.84*c

distance of earth from the mother ship=0.20602 light years

then time taken=0.20602 light years/(0.84*c)

as 1 light year=c*1 year

==>time taken=0.20602*c years/(0.84*c)=0.24527 years

part d:

w.r.t. earth based observer,

speed of the landing craft=0.95874*c

distance as observed from earth=0.26 light years

then time taken=distance/speed

=0.26*c years/(0.95874*c)

=0.27119 years