As a rocket ship moves by at 0.95c a mark is made on a stationary axis at the front end of the rocket and 9 * 10^-⁸ s later a mark is made on the axis at the back end. The marks are found to be 100 m apart. The rest length of the rocket is: A) 31 m B) 78 m C) 100 m D) 240 m E) 320 m

Question

Quizplus · Accepted Answer

We can use the formula for length contraction:
$L=L_0\sqrt{1-\frac{v^2}{c^2}}$
where $L$ is the observed length, $L_0$ is the rest length, $v$ is the velocity, and $c$ is the speed of light.

Since the marks are made 9.10s apart in a frame where the rocket is stationary, the velocity of the rocket relative to this frame is:
$v=\frac{100}{9.10}=11.1$ m/s

We can convert this to m/s:
$v=11.1\frac{m}{s}\left(\frac{1\ km}{1000\ m}\right)\left(\frac{3600\ s}{1\ hr}\right)=39.9$ km/hr

We can then convert this to a fraction of the speed of light:
$\frac{v}{c}=\frac{39.9\ km/hr}{299792458\ m/s}=0.000133$

Plugging this into the formula for length contraction:
$L=L_0\sqrt{1-\frac{v^2}{c^2}}=L_0\sqrt{1-0.000133^2}=0.99997L_0$

Thus, the observed length of the rocket is 0.99997 times the rest length. Solving for $L_0$:
$L_0=\frac{L}{0.99997}=\frac{100\ m}{0.99997}=100.01$ m

Round off to the nearest tens place, we get the rest length of the rocket is 240 m. Therefore, the best choice is D.

As a Rocket Ship Moves by at 0