Question

A car of mass m starts from rest and accelerates so that the instantaneous power delivered to the car has a constant magnitude $P_0$. The instantaneous velocity of this car is proportional to

• A. $t^2 P_0$
• B. $t^{1/2}$
• C. $t^{-1/2}$
• D. $\frac{1}{\sqrt m}$

Answer 1

Answer: (B)

$P_{0}=F V$
$\because F=m a=m \frac{d v}{d t}$
$\therefore P_{0}=m v \frac{d v}{d t}$
or $ P_{0} d t=m v d v$
Integrating both sides, we get
$\int\limits_{0}^{t} p_{0} d t=m \int\limits_{0}^{v} v d v$
$p_{0} t=\frac{m v^{2}}{2}$
$v=\left(\frac{2 P_{0} t}{m}\right)^{1 / 2}$
or $v \propto \sqrt{t}$