Question

At what speed, as a fraction of $c$, will a moving rod have a length$60\%$ that of an identical rod at rest?

Short answer

The speed is$0.8c$.

Step-by-step solution

Given Information

We need to find the peed, as a fraction of $c$, will a moving rod have a length $60\%$that of an identical rod at rest.

Simplify

The moving rod is the frame$S\text{'}$and the ground frame is $S$.The length of rod is $S\text{'}$is the proper length $l$because the rod is at rest is $S\text{'}$.

$L=\sqrt{1-{\beta }^2}l$

The length is calculated to $60\%$when

$$\begin{gathered} L=0.6l \\ \sqrt{1-{\beta }^2}=\sqrt{1-\frac{v^2}{c^2}}=0.6 \\ v=0.8c \end{gathered}$$