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

Explanation

As ` v= (dt) /(dt) = k sqrt x` ….(i)
or ` x^(-1//2) dx = k dt1
Integrating it wighin the conditions of motion (i.e. at ` t= 0, x=0 and at t=t , x= x), we have
` int _0 ^x x^(1/2) dx = int k dt`
or ` ` [( x(-1//2 + 1))/((- 1/2 +1))]_0 ^x = k (t) _0^t`
or ` 2 x^(1//2) = kt or x^(1//2) = kt //2`
Putting this value in (i), we get
` v= k ( kt//2) = k^@ t//2)`.