The displacement 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.

Options

A. varies with time
B. independent to time
C. inversely proportional to time
D. nothing can be said

Explanation

` v = k sqrt(x) rArr (dx)/(dt) = k sqrt(x) rArr int(dx)/(sqrt(x)) = int kdt rArr (x^(+1//2))/(1//2) = kt + c `
Given that, at t = 0, x = 0`therefore c = 0`
Now, ` 2x^(1//2) = kt rArr sqrt(x) = (1//2) kt `,
Now, ` v = k (1//2 kt ) = k^(2) t//2`
Thus velocity varies with time. Hence correct answer is (A)