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

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

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

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

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

answer is A.

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Detailed Solution

P = constant

$P = Fv = \frac{{mv}^{2} dv}{dx}$

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

$\frac{Px}{m} = \frac{v^{3}}{3}$

${(\frac{3 Px}{m})}^{\frac{1}{3}} = v = \frac{dx}{dt}$

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

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

Hence the correct answer is ${(\frac{8 P}{9 m})}^{\frac{1}{2}} t^{\frac{3}{2}} .$

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