An automobile of mass m accelerates starting from the origin and...

$P = const$

$P = {F}_{V} = \dfrac{m{v}^{2}dv}{dx}$

${∫}_{0}^{x}\dfrac{P}{m}dx = {∫}_{0}^{v}{v}^{2}dv$

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

\left(\dfrac{3Px}{m}\right)\dfrac{1}{3} = v = \dfrac{dx}{dt}$</p><p>\left(\dfrac{3P}{m}\right)\dfrac{1}{3}{∫}_{0}^{t}dt = {∫}_{0}^{x}{x}^{- \dfrac{1}{3}}dx$

$⇒ x = \left(\dfrac{8P}{9m}\right)\dfrac{1}{2}t\dfrac{3}{2}$