The dispkacement of particle is zero at ` t= 0` and it is ` x` , at ` t-t`. It starts moving in the positive x-direction with a velocity which varices as ` v= k sqrt x`, where (k) is a constant. Show that velocity is directly proprtional to time.

Explanation

Here, ` v=k sqrt x, (dx)/(dt) = k x^(1//2) ` or ` (dx)/x^(1//2) = k dt `
Integrating both sides within the linmits ` [ x=0` at
` t= 0 ` and ` x=x ` at ` t=t]`
` int _0^x ( dx)/x^(1//2) = int_0^t k dt` .
On soveing, ` sqrt x = ( kt) /2`
As ` `v=k sqrt x = k ((kt0/2) = (k^2 t)/2 ` :. ` v prop t` .