Question

Velocity of a bullet changes from $u$ to $v$ after passing through a board of thickness $d$. Force of resistance is directly proportional to the velocity. Time of motion of bullet in the board is given by

• A. $\frac{d(u-v)}{uv{\log }_e\frac{u}{v}}$
• B. $\frac{du}{v{\log }_e\frac{u}{v}}$
• C. $\frac{dv}{u{\log }_e\frac{u}{v}}$
• D. $\frac{d{\log }_e\frac{v}{u}}{v-u}$

Answer 1

Answer: (D)

$\frac{d{\log }_e\frac{v}{u}}{v-u}$

$\frac{dv}{dt}=\alpha v\Rightarrow \frac{dv}{v}=\alpha dt\Rightarrow ln\left(\frac{v}{u}\right)=\alpha t\frac{dv}{dt}=\alpha v\Rightarrow v\frac{dv}{dx}=\alpha v\Rightarrow v-u=\alpha d\Rightarrow \alpha =\frac{v-u}{d}\Rightarrow t=\frac{ln\left(\frac{v}{u}\right)}{\frac{v-u}{d}}=\frac{d\cdot ln\left(\frac{v}{u}\right)}{v-u}$