Question

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

Short answer

The length of the rocket $A$as measured by the crew of rocket $B$is $21.95m$.

Step-by-step solution

Given Information 

We need to find the length of rocket $A$ as measured by the crew of rocket $B$.

Simplify

Speed of rockets with respect to earth is $u=v=0.9c$. So the relative speed of the rocket with respect to each other is

$v_r=\frac{u+v}{1+{\displaystyle \frac{uv}{c^2}}}=\frac{0.8c+0.8c}{1+{\displaystyle \frac{0.8c\times 0.8c}{c^2}}}=0.97c$

Here,$L$is the length seen from each rocket to other rocket then we have

$L=\frac{l}{\gamma }=l\left(\sqrt{1-\frac{{v^2}_r}{c^2}}\right)$

At last, we know that $l=100m$. So

$L=100\times \sqrt{1-\frac{{\left(0.97c\right)}^2}{c^2}}=21.95m$