First slide

Instantaneous acceleration

Question

The deceleration experienced by a .moving motor boat, after its engine is cut off is given by (dv / dt)= - K v³ where K is a constant. If v_o is the magnitude of the velocity at cut off, the magnitude of the velocity at a time t after the cut-off is

Moderate

A

$v_{0} / 2$
B

v₀
C

v_oe^-Kt
D

$v_{0} / \sqrt{(2 v_{0}^{2} K t + 1)}$

Solution

Given,
$\frac{dv}{dt} = - {Kv}^{3} or \frac{dv}{v^{3}} = - Kdt$
$\begin{array}{l} ∴ \int_{v_{0} v^{3}}^{v} \frac{dv}{v} = - K \int_{v_{0}}^{t} dt \\ or {[- \frac{1}{2 v^{2}}]}_{v_{0}}^{v} = - K [t]_{0}^{t} \\ or [\frac{1}{2 v_{0}^{2}} - \frac{1}{2 v^{2}}] = - Kt \end{array}$

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