Question

Explain time dilation with an example.

Answer 1

Consider two frames $S$ and $S^{\prime }$. Let $S^{\prime }$ be moving with a velocity $v$ with respect to $S$ in the positive $X$-direction. Suppose a clock situated in the frame $S^{\prime }$ at a position gives out signals at an interval $t_0$.
If this interval is observed by an observer in frame $S$, then the interval $t$ recorded by him is,
$t=\frac{t_0}{\sqrt{1-\frac{v^2}{c^2}}}$ i.e. $t>t_0$
To a stationary observer in $S$, time interval appears to be lengthened by a factor $\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$. In other words, a moving clock appears to be slowed down to a stationary observer. This is known as time dilation.
Example: The clock in the moving space-ships will appear to go slower than the clocks on the earth.