Question

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

• A. $v_{0} c^{-k t}$
• B. $\frac{v_{0}}{\sqrt{2 v_{0}^{2} k t+1}}$
• C. $v_{0}$
• D. $\frac{v_{0}}{2}$

Answer 1

Answer: (B)

Here, $\frac{d v}{d t}=-k v^{3}$
or $\frac{d v}{v^{3}}=-k d t$
or $\int\limits_{v_{0}}^{v} \frac{d v}{v^{3}}=\int\limits_{0}^{t}-k d t$
or $\left[\frac{-1}{2 v^{2}}\right]_{v_{0}}^{v}=-k d t$
or $\frac{1}{2 v_{0}^{2}}-\frac{1}{2 v^{2}}=-k t$
or $\frac{1}{2 v^{2}}=\frac{1}{2 v^{2}}+k t$
or $\frac{1}{2 v^{2}}=\frac{1+2 v_{0}^{2} k t}{2 v_{0}^{2}}$
or $v^{2} =\frac{v_{0}^{2}}{2 v_{0}^{2} k t+1}$
or $v =\frac{v_{0}}{\sqrt{\left(2 v_{0}^{2} k t+1\right)}}$