First slide

Rectilinear Motion

Question

The retardation experienced by a moving motor boat, after its engine is cut-off, is given by $\frac{dv}{dt} = - {kv}^{3}$ where k is a constant. If v₀ is the magnitude of the velocity at cut-off, the magnitude of the velocity at time t after the cut-off is

Moderate

A

v₀
B

$\frac{v_{0}}{2}$
C

$v_{0} e^{- kt}$
D

$\frac{v_{0}}{\sqrt{2 v_{0}^{2} kt + 1}}$

Solution

$As, \frac{d v}{d t} = - k v^{3} \Rightarrow \int_{v_{0}}^{v} \frac{d v}{v^{3}} = - k \int_{0}^{t} d t \Rightarrow - \frac{1}{k} \int_{v_{0}}^{v} v^{- 3} d v = t o r - \frac{1}{k} {|\frac{v^{- 3 + 1}}{- 3 + 1}|}_{v_{0}}^{v} = t o r \frac{1}{2 k} [\frac{1}{v^{2}} - \frac{1}{v_{0}^{2}}] = t o r \frac{1}{v^{2}} - \frac{1}{v_{0}^{2}} = 2 k t o r \frac{1}{v^{2}} = \frac{1}{v_{0}^{2}} + 2 k t o r \frac{1}{v^{2}} = \frac{1 + 2 v_{0}^{2} k t}{v_{0}^{2}} o r v = \frac{v_{0}}{\sqrt{2 v_{0}^{2} k t + 1}}$

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