The velocity of particle moving in the positive direction of x axis varies as `v=alphasqrt(x)`, where `alpha` is a positive constnat. Assuming that at moment `t=0`, the particle was located at the point x=0. find.
(a) the time dependence of the velocity and the acceleration of the particle.
(b) 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)`