Answer
The length of Rocket A according to the crew of Rocket B is $~~22.0~m$
Work Step by Step
Let $u'$ be Rocket A's velocity relative to the earth reference frame.
Let $v$ be the earth's velocity relative to Rocket B.
Let $u$ be Rocket A's velocity relative to Rocket B.
We can find $u$:
$u = \frac{u'+v}{1+\frac{u'v}{c^2}}$
$u = \frac{0.800c+0.800c}{1+\frac{(0.800c)(0.800c)}{c^2}}$
$u = \frac{1.600c}{1+0.64}$
$u = 0.9756~c$
The speed of Rocket A relative to Rocket B is $~~0.9756~c$
We can find $\gamma$:
$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$
$\gamma = \frac{1}{\sqrt{1-\frac{(0.9756~c)^2}{c^2}}}$
$\gamma = \frac{1}{\sqrt{0.04820464}}$
$\gamma = 4.555$
We can find the length of Rocket A according to the crew of Rocket B:
$L = \frac{L_0}{\gamma}$
$L = \frac{100~m}{4.555}$
$L = 22.0~m$
The length of Rocket A according to the crew of Rocket B is $~~22.0~m$
