Question

A vehicle of mass $M$ is accelerated on a horizontal frictionless road under a force changing its velocity from $u$ to $v$ and distance covered is $S$. A constant power $P$ is given by the engine of the vehicle. The $v$ is equal to

• A. ${(u^3+\frac{2PS}{M})}^{1/3}$
• B. ${(\frac{PS}{M}+u^3)}^{1/2}$
• C. ${(\frac{PS}{M}+u^2)}^{1/3}$
• D. ${(\frac{3PS}{M}+u^3)}^{1/3}$

Answer 1

The correct option is D ${(\frac{3PS}{M}+u^3)}^{1/3}$

We know that,
P=Fv=mav
$\Rightarrow P=mv\frac{dv}{dt}$
$\Rightarrow P=mv\frac{dv}{dx}\frac{dx}{d}=m{v}^2\frac{dv}{dx}$
$\Rightarrow Pdx=M{v}^{2}dv$
$\Rightarrow {\int }_0^{s}Pdx=M{\int }_u^v{v}^{2}dv$
$\Rightarrow PS=M\frac{(v^3-u^3)}{3}$
$\Rightarrow v=(\frac{3PS}{M}+u^3{)}^{\frac{1}{3}}$