Question

Two rockets, A and B, approach the earth from opposite directions at speed 0.80c. The length of each rocket measured in its rest frame is 100 m. What is the length of rocket A as measured by the crew of rocket B?

Answer 1

Answer:

The contracted length as measured is 22.22 m

Solution:

As per the question:

Speed of the rockets, v = 0.8c

Length of the rocket, L =  100 m

Now,

To calculate the length of rocket A w.r.t rocket B:

The relative speed of the rockets A and B approaching the earth from the opposite directions:

$Relative speed,\ v_{r} = \frac{- 0.8c - 0.8c}{1 + \frac{v^{2}}{c^{2}}}$

$v_{r} = \frac{- 1.6c}{1 + \frac{(0.8c)^{2}}{c^{2}}} = 0.975c$

The contracted length is given by:

$L' = \frac{L}{\sqrt{1 - \frac{v_{r}^{2}}{c^{2}}}}$

$L' = \frac{100}{\sqrt{1 - \frac{(0.975c)^{2}}{c^{2}}}} = 22.22\ m$