Question

A machine delivers power to a body which is proportional to instantaneous velocity $v$ of the body. If the body starts with a velocity which is almost negligible, then distance covered by the body is proportional to

• A. $\sqrt{v}$
• B. $\sqrt[3]{3\frac{v}{2}}$
• C. $v^{5/3}$
• D. $v^2$

Answer 1

The correct option is D $v^2$

Power $=Fv=m\frac{dv}{dt}\left(v\right)\propto v$

$⟹mv\frac{dv}{dt}=k_{0}v⟹mv\frac{dv}{dx}=k_0$

$⟹vdv=\frac{k_0}{m}dx$

${\int }_0^{v}vdvdv=\frac{k_0}{m}{\int }_0^{x}dx$

$=\frac{v^2}{2}=\frac{k_0}{m}x$

Hence, displacement is proportional to square of instantaneous velocity.