A spaceship passes you at a speed of 0.850c. You measure its length to be 48.2m. How long would it be when at rest?

Question

theeric · Answer

Hello! This covers the "length contraction" phenomena.

You start out knowing the relative speed between you and it. That is $.850c$ and we'll call that $v$.

You measure its length to be $48.2\ [m]$ when it's moving that fast. But, at relative velocity near the speed of light, it has been contracted. We'll call that contracted length $L$.

So the real length must actually be longer.

The formula you must use is $$L_0=L\ \gamma$$
The $\gamma$ is the Lorentz factor, which is important in many equations about special relativity. It's much easier to look at that than what it's equal to. $\gamma=\dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}$. In that, $c$ is the speed of light, and $v$ is the relative velocity we mentioned before.

Now you have $L_0=L\ \dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}$. You just have to plug the values in, now!