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 $${v_0} = 10\,m{s^{ - 1}}.$$ If, after $$10 \,s,$$ its energy is $$\frac{1}{8}mv_0^2,$$ the value of $$k$$ will be-
A.
$${10^{ - 4}}\,kg\,{m^{ - 1}}$$
B.
$${10^{ - 1}}\,kg\,{m^{ - 1}}{s^{ - 1}}$$
C.
$${10^{ - 3}}\,kg\,{m^{ - 1}}$$
D.
$${10^{ - 3}}\,kg\,{s^{ - 1}}$$
Answer :
$${10^{ - 4}}\,kg\,{m^{ - 1}}$$
Solution :
Let $${V_f}$$ is the final speed of the body.
From questions,
$$\eqalign{
& \frac{1}{2}mV_f^2 = \frac{1}{8}mV_0^2\,\,\,\,\, \Rightarrow {V_f} = \frac{{{V_0}}}{2} = 5\,m/s \cr
& F = m\left( {\frac{{dV}}{{dt}}} \right) = - k{V^2}\,\,\,\,\therefore \left( {{{10}^{ - 2}}} \right)\frac{{dV}}{{dt}} = - k{V^2} \cr
& \int\limits_{10}^5 {\frac{{dV}}{{{V^2}}}} = - 100K\int\limits_0^{10} {dt} \cr
& \frac{1}{5} - \frac{1}{{10}} = 100\,K\left( {10} \right) \cr
& {\text{or}},\,\,K = {10^{ - 4}}\,kg\,{m^{ - 1}} \cr} $$