A car of mass m starts from rest and accelerates so that the instantaneous power delivered to the car has a constant magnitude P0. The instantaneous velocity of this car is proportional to

Question

A car of mass m starts from rest and accelerates so that the instantaneous power delivered to the car has a constant magnitude P0. The instantaneous velocity of this car is proportional to (A) t2 P0 (B) t1/2 (C) t-1/2 (D) (1/√ m)

Tardigrade Admin · Accepted Answer

P0=F V ∵ F=m a=m (d v/d t) ∴ P0=m v (d v/d t) or P0 d t=m v d v Integrating both sides, we get ∫ limits0t p0 d t=m ∫ limits0v v d v p0 t=(m v2/2) v=((2 P0 t/m))1 / 2 or v propto √t

Q. 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

$t^{2} P_{0}$

$t^{1/2}$

$t^{- 1/2}$

$\frac{1}{m}$

$P_{0} = F V$
$∵ F = ma = m \frac{d v}{d t}$
$∴ P_{0} = m v \frac{d v}{d t}$
or $P_{0} d t = m v d v$
Integrating both sides, we get
$0 \int t p_{0} d t = m 0 \int v v d v$
$p_{0} t = \frac{m v ^{2}}{2}$
$v = (\frac{2 P _{0} t}{m})^{1/2}$
or $v \propto t$

Q. A car of mass m starts from rest and accelerates so that the instantaneous power delivered to the car has a constant magnitude P0​. The instantaneous velocity of this car is proportional to

t2P0​

t1/2

t−1/2

m​1​