The velocity of a particle moving in the positive direction of the x axis varies as `v=alphasqrtx`, where `alpha` is a positive constant. Assuming that at the 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 the first s metres of the path.

Explanation

(a) `v =alpha sqrtx rArr (dx)/(dt) = alpha sqrtx rArr overset x underset0 int (dx)/(sqrtx) = alpha overset t underset0 int dt rArr (x^((1)/(2)+1))/(-(1)/(2)+1)= alpha t rArr x = (alpha^(2) t^(2))/(4)`
Velocity v `=(dx)/(dt) = ( 2alpha^(2)t)/(4) = (1)/(2) alpha^(2)t`
Acceleration `a =(dv)/(dt) = (alpha^(2))/(2)`
(b) Time take to cover the first s distance
`x = (alpha^(2)t^(2))/(4) rArr s = (alpha^(2)t^(2))/(4) rArr v_(av) = (s)/(t) = (s)/(sqrt((4s)/(alpha^(2))) = (alpha sqrts)/(2)`