Question

You have been posted to a remote region of space to monitor traffic. Near the end of a quiet shift, a spacecraft streaks past. Your laser-based measuring device reports the spacecraft’s length to be 85 m. The identification transponder reports it to be the NCXXB12, a cargo craft of proper length 100 m. In transmitting your report to headquarters, what speed should you give for this spacecraft?

Answer 1

Answer:

The value is $v =1.58 *10^{8} \ m/s$

Explanation:

From the question we are told that

The length of the spacecraft streaks is $L_i = 85 \ m$

The proper length of the spacecraft streaks $L_p = 100 \ m$

Generally from the equation of relativity of length contraction

$L_i = L_p * \sqrt{1 - \frac{v^2}{c^2} }$

Here c is the speed of light with value $c = 3.0 *10^{8} \ m/s^2$

v is the velocity of the spacecraft

So

$\frac{v^2}{c^2} = 1 - \frac{L_i^2}{L_p^2 }$

=> $\frac{v^2}{c^2} = \frac{L_p^2 - L_i^2}{L_p^2}$

=> $\frac{v^2}{c^2} = \frac{100^2 - 85^2}{100^2}$

=> $\frac{v^2}{c^2} = 0.2775$

=>  $v = \sqrt{0.2775c^2}$

=>  $v = 0.5268c$

=>  $v = 0.5268 * 3.0*10^{8}$

=>  $v =1.58 *10^{8} \ m/s$