# Finding the distance travelled during change in velocity from $v_i$ to $v_f$ [closed]

A constant power is applied to a particle of mass m. The velocity of the particle increases from $v_i$ to $v_f$. We have to find the distance travelled by the particle during this interval.(neglect friction)

$$P=Fv_f$$ $$P=mav_f$$ $$P=\frac{mv_f(v_f^2-v_i^2)}{2S}$$ $$S=\frac{mv_f(v_f^2-v_i^2)}{2P}$$

where

P=>Power

F=>Force

m=>Mass

S=>Distance

However the correct answer is supposed to be $\frac{m(v_f^3-v_i^3)}{3P}$. Can anyone explain this.

Using Chain rule, We get $a= \frac{vdv}{dx}$,
$$P=constant\\ P=F.v\\ P=(ma).v\\ \\ P=\left(m\frac{v\space dv}{dx}\right).v$$ Now Integrate,