A point moves rectilinearly with deceleration whose modulus depends on the velocity v of the paticles as `alpha=ksqrt(v)`, where k is a positive constant .At the initial movement the velocity of the point in equal to `V_(0)`. What distance will it take to cover that distance?

Explanation

Let `t_(0)` be the time which it come to a stop.
Given that `-(dt)/(dt)=ksqrt(v)" "underset(0)overset(t_(0))int kdt=underset(v_(0))overset(0)int-(dv)/(sqrt(V))`
`because kt_(0)=2sqrt(v_(0)) therefore=(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)=-Ksqrt(V),`
`therefore v(dv)/(ds)=-ksqrt(V) therefore sqrt(dv)=-kds`
`therefore underset(v_(0))overset(0)intsqrt(v_(0))int sqrt(v)dv=-underset(0)overset(s)int` k ds`implies s(2)/(3k)V_(0)^((3)/(2))`