Question

Having gone through a plank of thickness $h$, a bullet changed its speed from $V_0$ to $\frac{V_0}{2}$. Find the time of motion of the bullet in the plank assuming the resistant force to be proportional to the speed.

• A. $\frac{2hln2}{V_0}$
• B. $\frac{hln2}{V_0}$
• C. $\frac{2h}{V_0}$
• D. None of these

Answer 1

The correct option is A $\frac{2hln2}{V_0}$

Let us assume that bullet covers the distance $h$ in time $t$.


Given,
$F\propto V$
$\Rightarrow F_R=-k^{{}^{\prime }}V$ (Here, -ve sign indicates that the force is resistive in nature and $k^{{}^{\prime }}$ is any constant)
Or, $a=\frac{-k^{{}^{\prime }}}{m}V=-kV$
$a=-kV$
$\begin{array}{cc}V\frac{dV}{dx}=-kV& a=-kV\\ dV=-kdx& \frac{dV}{dt}=-kV\\ {\int }_{V_0}^{\frac{V_0}{2}}dV=-k{\int }_0^{h}dx& {\int }_{V_0}^{\frac{V_0}{2}}\frac{dV}{v}=-k{\int }_0^{t}dt\\ -\frac{V_0}{2}=-kh& ln\frac{V_0}{2}-ln{V}_0=-kt\\ k=\frac{V_0}{2h}\dots \dots \left(1\right)& t=\frac{ln2}{k}\dots \dots \left(2\right)\\ & \text{From (1) and (2)}\\ & t=\frac{2hln2}{V_0}\end{array}$