The displacement to particle is zero at ` t=0` . It starts moving in the positive x-direction with a velocity which varies, `v= k sqrt x`, where (k) is a constant. Find the relation for variation of velocity with time.

Options

A. (a) (2 k^2)/t`
B. (b) (2 t^2)/k`
C. (c ) ` (k^2 t)/2`
D. (d) `2/(kt^2)`

Answer

C

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`