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

The seed as a fraction of the speed of light is
`(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 =
`(5)/(4)t_(i)`

Now,

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

`t = \frac{t_(i)}{\sqrt{1 - (v^(2))/(c^(2))}}`

`(5)/(4)t_(i) = \frac{t_(i)}{\sqrt{1 - (v^(2))/(c^(2))}}`

Squaring both sides:

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

`(25)/(16) = \frac{1}{{1 - (v^(2))/(c^(2))}}`

`1 - (16)/(25) = (v^(2))/(c^(2))`

`(v)/(c) = \sqrt{(9)/(25)}`

`(v)/(c) = (3)/(5)`

`v = (3)/(5)c`