First slide

Instantaneous acceleration

Question

The deceleration experienced by a moving motorboat, after its engine is cut-off is given by $\frac{dv}{dt} = - {kv}^{3}$ where k is constant. If v_o is the magnitude of velocity at cut-off, the magnitude of velocity at time 't' after the cut-off is:

Moderate

A

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

v_o
C

v_oe^-kt
D

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

Solution

$\frac{dv}{dt} = - {kv}^{3} \Rightarrow \int_{v_{o}}^{v} v^{- 3} dv = \int_{0}^{t} k dt \Rightarrow {(\frac{v^{- 2}}{- 2})}_{v_{o}}^{v} = - kt$
$\Rightarrow - \frac{1}{2} [\frac{1}{v^{2}} - \frac{1}{v_{0}^{2}}] = - kt$
$\Rightarrow \frac{1}{v^{2}} = \frac{1}{v_{0}^{2}} + 2 kt$
$\Rightarrow v = \frac{v_{o}}{\sqrt{2 v_{0}^{2} kt + 1}}$

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