Q77 / 90 · JEE Main 2021 (27 Jul Shift 2) · Physics

An automobile of mass $m$ accelerates starting from the origin and initially at rest, while the engine supplies constant power $P$ . The position is given as a function of time by:

Solution

$P = const$

$P = F_{V} = \frac{m v^{2} d v}{d x}$

$\int_{0}^{x} \frac{P}{m} dx = \int_{0}^{v} v^{2} dv$

$\frac{P x}{m} = \frac{v^{3}}{3}$

${(\frac{3 P x}{m})}^{\frac{1}{3}} = v = \frac{d x}{d t}$

${(\frac{3 P}{m})}^{\frac{1}{3}} \int_{0}^{t} d t = \int_{0}^{x} x^{- \frac{1}{3}} d x$

$\Rightarrow x = {(\frac{8 P}{9 m})}^{\frac{1}{2}} t^{\frac{3}{2}}$

Hence, the correct answer is option: 'D'.