Question

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

Answer 1

Answer:

The seed as a fraction of the speed of light is $\frac{3}{5}c$

Solution:

As per the question:

Suppose, $t_{i}$ be the rate of an identical clock between two time intervals.

For a moving clock, moving with velocity 'v', at the clock tick of four-fifth:

t = $\frac{5}{4}t_{i}$

Now,

Using the relation of time dilation, from Einstein's relation:

$t = \frac{t_{i}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}$

$\frac{5}{4}t_{i} = \frac{t_{i}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}$

Squaring both sides:

$(\frac{5}{4})^{2} = (\frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}})^{2}$

$\frac{25}{16} = \frac{1}{{1 - \frac{v^{2}}{c^{2}}}}$

$1 - \frac{16}{25} = \frac{v^{2}}{c^{2}}$

$\frac{v}{c} = \sqrt{\frac{9}{25}}$

$\frac{v}{c} = \frac{3}{5}$

$v = \frac{3}{5}c$