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 $v_0=10m/s$. If, after $10s$, its energy is $\frac{1}{8}m{v}_0^2$, the value of $k$ will be

The correct option is A ${10}^{-4}kg/m$
Step1: Find the final velocity of the body.

Given that,
Mass $m={10}^{2}kg$
Initial velocity $v_0=10m/s$
Time $t=10s$
Frictional force $F=-K{v}^2$
Now,
$\frac{1}{2}m{v}_f^2=\frac{1}{8}m{v}_0^2$

$v_f=\frac{v_0}{2}$

$v_f=\frac{10}{2}$

$=5m/s$

Step2: Find the value of $k$.
Formula Used: $F=ma$
$-k{v}^2=\frac{mdv}{dt}$

${\int }_0^{\frac{v_0}{2}}\frac{dv}{v^2}={\int }_0^{t_0}\frac{-kdt}{m}$

${[-\frac{1}{v}]}_{v_0}^{\frac{v_0}{2}}=\frac{-k}{m}{t}_0$

On solving we get
$k=\frac{m}{v_0{t}_0}$

$=\frac{{10}^{-2}}{10\times 10}={10}^{-4}kg/m$

Final Answer: (a)