The velocity of a particle moving in the positive direction of `x`-axis varies as `v = alpha sqrt(x)` where `alpha` is positive constant. Assuming that at the moment `t = 0`, the particle was located at `x = 0`, find (i) the time dependance of the velocity and the acceleration of the particle and (ii) the mean velocity of the particle averaged over the time that the particle takes to cover first `s` meters of the path.

Explanation

(i) Given, the velocity of the particle moving in the positive direction of X-axis.
`v = alpha sqrt(x) rArr (dx)/(dt) = alpha sqrt(x)`
By integrating `int_(0)^(x)(dx)/(sqrt(x))=alpha int_(0)^(x)dt`
`rArr (x^(-1//2+1))/(-(1//2)+1)=alpha t rArr x=(alpha^(2)t^(2))/(4)`
Time dependence of the velocity of the particle
`v=(dx)/(dt)=(2alpha^(2)t)/(4)=(1)/(2)alpha^(2)t`
(ii) Distance, `x = (alpha^(2)t^(2))/(4)`' for s distance `s = (alpha^(2)t^(2))/(4)`
Time taken to cover first s distance, `t = sqrt((4s)/(alpha^(2)))`
The mean velocity of the particle,
`v_(Av)=(s)/(t)=(s)/(sqrt(4s//alpha^(2)))=(alpha sqrt(s))/(2)`