A car of mass $m$ starts from rest and accelerates so that the instantaneous power delivered to the car has…
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
$t^2 P_0$
$t^{1 / 2}$
$t^{-1 / 2}$
$t / \sqrt{m}$
Solution
Power $P_0=F v$
$\begin{aligned}
& =\left(m \frac{d v}{d t}\right) v \\
& =m v \frac{d v}{d t}
\end{aligned}$
Integrating both sides
$\begin{aligned}
\int P_0 d t & =\int m v d v \\
P_0 t & =\frac{m v^2}{2} \\
v^2 & =\frac{2 P_0 t}{m} \\
v & \propto t^{1 / 2}
\end{aligned}$