Question
A car of mass $$m$$ starts from vest 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.
$${t^{\frac{1}{2}}}$$
C.
$${t^{ - \frac{1}{2}}}$$
D.
$$\frac{t}{{\sqrt m }}$$
Answer :
$${t^{\frac{1}{2}}}$$
Solution :
Constant power of car $${P_0} = F.V = ma.v$$
$$\eqalign{
& {P_0} = m\frac{{dv}}{{dt}} \cdot v \cr
& {P_0}dt = mvdv\,{\text{Integrating}} \cr
& {P_0}.t = \frac{{m{v^2}}}{2}\,\,v = \sqrt {\frac{{2{P_0}t}}{m}} \cr
& \because {P_0},m\,{\text{and}}\,2\,{\text{are}}\,{\text{constant}} \cr
& \therefore v \propto \sqrt t \cr} $$