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Q.

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 $v_{0} = 10 m s^{- 1}$ . If after 10s, its energy $\frac{1}{8} m v_{0}^{2}$ is , the value of k will be:

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a

$10^{- 3} {kg m}^{- 1}$

b

$10^{- 1} {kg m}^{- 1} s^{- 1}$

c

$10^{- 3} kg s^{- 1}$

d

$10^{- 4} {kg m}^{- 1}$

answer is A.

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Detailed Solution

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Let $V_{f}$ is the final speed of the body. From questions, $\frac{1}{2} m V_{f}^{2} = \frac{1}{8} m V_{0}^{2}$ , $\Rightarrow V_{f} = \frac{V_{0}}{2} = 5 m / s$

$\begin{array}{l} F = m (\frac{d V}{d t}) = - k V^{2} ∴ (10^{- 2}) \frac{d V}{d t} = - k V^{2} \\ \int_{10}^{5} \frac{d V}{V^{2}} = - 100 K \int_{0}^{10} d t, \frac{1}{5} - \frac{1}{10} = 100 K (10) o r, K = 10^{- 4} k g m^{- 1} \end{array}$

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