Question

Two identical spaceships, both measuring 100 m at rest are moving in the same direction. In a particular reference frame one spaceship measures 80 m, and the other 60 m. What would be the lengths of the spaceships with respect to each other?

Answer 1

Solution:

Length of the spaceship when they are at rest = L_o = 100 m (original length )

Relative length of the spaceships: L1 =80 m ; L2 =60m

According to special relativity, L= Lo [ 1 -v²/c² ]^1/2

length contraction occurs when the moving spaceshipis observed by an observer in a reference frame at rest .

80 = 100(1-v₁²/c² )^1/2

and 60 = 100 ( 1-v₂²/c²)^1/2

=> v1 = 0.6c and v2 = 0.8c

From the reference frame referred, these are their respective speeds.

Speed of 1 as seen by 2 = (v1+v2) /(1+v2v1/c² )

=(0.6c +0.8c) / [1+(0.6*0.8)c^2/c^2 ] = 0.095 c

Speed of 2 as seen by 1 = (0. 8+0.6)c /(1+0.48c^2/c^2) = 0.095c

Length of 1 as seen by 2 = L1 = 100 [(1-0.095 c^2/c^2 ]^1/2

                                                 = 99.5 m

Length of 2 as seen by 1 = 60[(1-0.095 c^2/c^2 ]^1/2

                                        = 100( 0.995) = 99.5 m