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

(i) Given that `v - alpha sqrt(x)` or `(dx)/(dt) = alpha sqrt(x)`
`:. (dx)/(sqrt(x)) = alpha dt` or `int_(0)^(x) (dx)/(sqrt(x)) = int_(0)^(t) alpha dt`
Hence `2sqrt(x) = alpha t`or `x = (alpha^(2)t^(2)//4)`
Velocity `(dx)/(dt) = (1)/(2)alpha^(2)t` and
Acceleration `(d^(2)x)/(dt^(2)) = (1)/(2)alpha^(2)`
(ii) Time taken to cover first `s` meters
`s = (alpha^(2)t^(2))/(4) rArr t^(2) = (4s)/(alpha^(2)) rArr t = (2sqrt(s))/(alpha)`
`barv = ("total distance")/("total time") = (sa)/(2sqrt(s)) rArr barv = (1)/(2) sqrt(s) alpha`