Question

Two rockets are each $1000m$long in their rest frame. Rocket Orion, traveling at $0.80c$ relative to the earth, is overtaking rocket Sirius, which is poking along at a mere $0.60c.$ According to the crew on Sirius, how long does Orion take to completely pass? That is, how long is it from the instant the nose of Orion is at the tail of Sirius until the tail of Orion is at the nose of Sirius?

Short answer

The time is $1.6\times {10}^{-5}s$from the instant the nose of Orion is at the tail of Sirius until the tail of Orion is at the nose of Sirius.

Step-by-step solution

Given Information

We need to find that the from the instant the nose of Orion is at the tail of Sirius until the tail of Orion is at the nose of Sirius

Simplify

The relative between the rocket is :

$v_r=\frac{v-u}{1-{\displaystyle \frac{uv}{c^2}}}=\frac{0.8c-0.6c}{1-{\displaystyle \frac{0.8c\times 0.6c}{c^2}}}=0.38c$

Therefore the length of the Orion seen from the Sirius is :

$L=l\sqrt{1-\frac{v^2}{c^2}}=1000\times \sqrt{1-\frac{{\left(0.38c\right)}^2}{c^2}}=855.6$

So total length that Orion has to cross is $1000+844.6=1855.6m$with speed $v_r=0.38\times 3\times {10}^{8}m/s$

here, time taken by Orion seen be the crew at Sirius is:

$t=\frac{1855.6}{0.33\times 3\times {10}^8}=1.6\times {10}^{-5}s$