Physics - W.E.P. Question with Solution | TestHub

Power $P=\frac{dW}{dt}$. Since the road is frictionless and horizontal, all the work done by the engine goes into changing the kinetic energy of the car. So, $P=\frac{d(KE)}{dt}$. Integrating this with respect to time, $KE=Pt+C$. Since the car starts from rest, at $t=0$, $KE=0$, so the integration constant $C=0$. Thus, $KE=Pt$. This means the kinetic energy of the car increases linearly with time, so Reason (R) is true. Also, we know that $KE=\frac{1}{2}m{v}^2$. Substituting $KE=Pt$, we get $\frac{1}{2}m{v}^2=Pt$, which implies $v^2=\frac{2Pt}{m}$. Taking the square root of both sides, $v=\sqrt{\frac{2P}{m}}{t}^{1/2}$. Hence, $v\propto t^{1/2}$, so Assertion (A) is true. Reason (R) correctly explains Assertion (A) because the linear increase of kinetic energy with time directly leads to the $t^{1/2}$ dependence of velocity.