Question

At what speed, as a fraction of $c,$does a moving clock tick at half the rate of an identical clock at rest?

Short answer

At $0.866c$, a moving clock tick at half the rate of an identical clock at rest.

Step-by-step solution

Given Information

We have to find the speed of the moving clock as a fraction of$c.$

Simplify

The ticks that are being counted are those of a ticking clock. In $S^{\text{'}}-$the time interval between two ticks is measured by the same clock, namely the ticking clock. As a result, $∆t\text{'}=∆\tau$is the correct time. Between two ticks of the running clock, the rest clock measures a longer interval $∆t$. These are linked by the following:

$∆\tau =\sqrt{1-{\beta }^2}∆t$

Now $∆\tau =\frac{1}{2}∆t,$Thus,

$\sqrt{1-{\beta }^2}=\sqrt{1-\frac{v^2}{c^2}}=\frac{1}{2}$

$$\begin{gathered} v=c\sqrt{\frac{3}{4}} \\ v=0.866c \end{gathered}$$