The retardation experienced by a moving motorboat, after its engine is cut off, is given by $\frac{d v}{d t} = - k v^{3},$ where $k$ is a constant. If $v_{0}$ is the magnitude of the velocity at cut-off, the magnitude of the velocity at time $t$ after the cut-off is

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d v d t = - k v 3 ⇒ - ∫ v 0 v d v k v 3 = ∫ 0 t d t ⇒ 1 k ∫ v 0 v - v - 3 d v = t or - 1 k v - 3 + 1 - 3 + 1 v 0 v = t or 1 2 k 1 v 2 - 1 v 0 2 = t or 1 v 2 - 1 v 0 2 = 2 k t or 1 v 2 = 1 v 0 2 + 2 k t or 1 v 2 = 1 + 2 v 0 2 k t v 0 2 or v = v 0 2 v 0 2 k t + 1

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