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A body of mass $m= 10^{-2} \;kg$ is moving in a medium and experiences a frictional force $F= -kv^2$ Its intial speed is $v_0= 10$ $ms^{-1}$ If, after $10\ s$, its energy is $\frac{1}{8}$ $mv_0^2$ the value of $k$ will be
$10^{-3} $ $kg m^{-1}$
$10^{-3}$ $kg s^{-1}$
$10^{-4}$ $kg m^{-1}$
$10^{-1}$ $kg m^{-1} s^{-1}$
Solution
Let ${V_f}$ is the final speed of the body.
From questions,
$\begin{gathered}
\frac{1}{2}mV_f^2 = \frac{1}{8}mV_0^{2\,}\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,{V_f} = \frac{{{V_0}}}{2} = 5m/s \hfill \\
F = m\left( {\frac{{dV}}{{dt}}} \right) = – k{V^2}\,\,\therefore \left( {{{10}^{ – 2}}} \right)\frac{{dV}}{{dt}} = – k{V^2} \hfill \\
\int\limits_{10}^5 {\frac{{dV}}{{{V^2}}} = – 100K\int\limits_0^{10} {dt} } \hfill \\
\frac{1}{5} – \frac{1}{{10}} = 100k\left( {10} \right)\,\,\,\,or,\,\,K = {10^{ – 4}}\,kg{m^{ – 1}} \hfill \\
\end{gathered} $
