A point moves rectilinearly with deceleration whose modulus depends on the velocity v of the particle as `alpha=ksqrtv`, where k is a positive constant. At the initial moment the velocity of the point is equal to `V_0`. What distance will it take to cover that distance?

Explanation

Let `t_(0)` be the time in which it comes to a stop.
Given that `-(dv)/(dt)=k sqrtV" "underset(0)overset(t_(0))int kdt=underset(v_(0))overset(0)int-(dv)/sqrtv`
`kt_(0)=2sqrt(v_(0))" "t_(0)=2/k sqrt(v_(0))`
Now to find the distance covered before stopping,
`(dv)/(dt)=(dv)/(ds) (ds)/(dt)=v""(dv)/(ds)," But "(dv)/(dt)=-k sqrt(V),`
`v".(dv)/(ds)=-k sqrt(V)" "sqrtv dv=-kds`
`underset(v_(0))overset(0)int sqrtv dv=-underset(0)overset(s)int k ds rArr s=2/(3k) V_(0)^(3/2)`