Question

A body of mass $m=10^{- 2 }kg$ is moving in a medium and experiences a frictional force $F = - k v^{2} .$ Its initial speed is $v_{0}=10ms^{- 1}.$ After $10s$ its kinetic energy is $\frac{1}{8}mv_{0}^{2},$ then value of $k$ will be:-

• A. $10^{- 1}kgm^{- 1}s^{- 1}$
• B. $10^{- 3}kgm^{- 1}$
• C. $10^{- 3}kgs^{- 1}$
• D. $10^{- 4}kgm^{- 1}$

Answer 1

Answer: (D)

After $10 \, \text{sec,}$ $\frac{1}{2}mv^{2}=\frac{1}{8}mv_{0}^{2}$
$\Rightarrow v=\frac{v_{0}}{2}=5ms^{- 1}$
Acceleration, $a = - \frac{k v^{2}}{m}$
$\Rightarrow \frac{d v}{d t} = - \frac{k v^{2}}{m}$
$\Rightarrow \displaystyle \int _{10}^{5}\frac{d v}{v^{2}}=-\frac{k}{m}\displaystyle \int _{0}^{10}dt$
$-\frac{1}{5}+\frac{1}{10}=\frac{- k}{10^{- 2}}\times 10$
$k=10^{- 4}kgm^{- 1}$