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?

Short answer

The length of rocket A as measured by the crew of rocket B 22 m.

Step-by-step solution

Given information

We have given that, 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.

Step 1.

Let us assume that, S be the earth’s frame of reference, and S′ be the rocket B's frame of reference. S′ moves with a velocity v = −0.8 c relative to S. In S, rocket A has a speed u = 0.8c.

Step 2. 

Using the Lorentz velocity transformation equation to calculate the speed of rocket A as observed from rocket B :

$u\text{'}=\frac{u-v}{1-{\displaystyle \raisebox{1ex}{$uv$}\!\left/ \!\raisebox{-1ex}{$c^2$}\right.}}$

$u\text{'}=\frac{0.8c-(-0.8c)}{1-{\displaystyle \raisebox{1ex}{$(0.8c)(-0.8c)$}\!\left/ \!\raisebox{-1ex}{$c^2$}\right.}}$

$u\text{'}=0.976c$

The length of rocket A from rocket B is :

$L=(100m)\sqrt{1-{(0.976)}^2}$

$L=22m$