Question

A comet is observed to be 130 km long as it moves past an observer at a speed of 0.700c. How long does the comet seem when it travels at a speed of 0.900c with respect to the observer?

Answer 1

When comet moves with the speed $v_1=0.7c$, it's length:

$l_1=l_0\sqrt{1-(\frac{v_1}{c})^2}$

When it moves with the speed $v_2=0.9c$:

$l_2=l_0\sqrt{1-(\frac{v_2}{c})^2}$.

We have the system of equation, where we must find $l_2$. Rewrite it in another form.

$l_1=l_0\sqrt{\frac{c^2-{v_1}^2}{c^2}}$

$l_2=l_0\sqrt{\frac{c^2-{v_2}^2}{c^2}}$

Divide these equations.

$\frac{l_2}{l_1}=\sqrt{\frac{c^2-{v_2}^2}{c^2-{v_1}^2}}$

Substitute numbers.

$\frac{l_2}{l_1}=\sqrt{\frac{c^2-0.81c^2}{c^2-0.49c^2}}=\sqrt{\frac{0.19}{0.51}}=0.61$

$l_2=0.61l_1=0.61*130=79$ km.