The speed of car of mass m starting from rest moving in a straight line under the influence of constant power P is given by $v=k (\frac{Px}{m})^{\frac {1}{3}}$ after travelling distance x ignoring energy losses. k is equal to,

(A) 2
(B) $2^\frac {1}{3}$
(C) 3
(D) $3^\frac {1}{3}$

Solution

$P=Fv=m\frac {dv}{dt} .v$

$\therefore P=m\frac {dv}{dx} . \frac {dx}{dt} .v = mv^2 \frac {dv}{dx}$

$\therefore \int\limits_0^x {Pdx} = \int\limits_0^v {m{v^2}dv} $

$\therefore Px = m \frac {v^3}{3} $

$\Rightarrow v=(\frac {3Px}{m})^{\frac {1}{3}}$

$\therefore k = 3^{\frac {1}{3}}$

Hence, (D)