Question

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

• A. $10^{- 1}k\text{g }\text{m}^{- 1}\text{s}^{- 1}$
• B. $10^{- 3}\text{ kg }\text{m}^{- 1}$
• C. $10^{- 3} \, \text{kg s}^{- 1}$
• D. $10^{- 4} \, \text{kg} \, \text{m}^{- 1}$

Answer 1

Answer: (D)

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