Question

With what velocity (relative to the reference frame $K$) did the clock move, if during the time interval $t=5.0s$, measured by the clock of the frame $K$, it became slow by $\Delta t=0.10s$?

Answer 1

Because of time dilation, a moving clock reads less time. We write,
$t-\Delta t=t\sqrt{1-{\beta }^2}$, $\beta =\frac{v}{c}$
Thus, $1-\frac{2\Delta t}{t}+{\left(\frac{\Delta t}{t}\right)}^2=1-{\beta }^2$
or, $v=c\sqrt{\frac{\Delta t}{t}(2-\frac{\Delta t}{t})}$

$⟹v=0.199c$