A body of mass \(m= 10^{-2}\) \(kg\) is moving in a medium and experiences a frictional force \(F= -kv^2\). Its initial 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

Now, \(\frac{\mathrm{mdv}}{\mathrm{dt}}=-\mathrm{kv}^{2}\)

\(\Rightarrow \mathrm{m} \int_{\mathrm{r}_{0}}^{\frac{\mathrm{r}_{0}}{2}} \frac{\mathrm{dv}}{\mathrm{v}^{2}}=-\mathrm{k} \int_{0}^{10} \mathrm{dt}\)

\(\Rightarrow \mathrm{m}\left[-\frac{1}{\mathrm{v}}\right]_{-\mathrm{o}}^{\frac{v_{0}}{2}}=-\mathrm{k}[\mathrm{t}]_{0}^{10}\)

\(\Rightarrow \mathrm{m}\left(\frac{2}{\mathrm{v}_{0}}-\frac{1}{\mathrm{v}_{0}}\right)=10 \mathrm{k}\)

\(\Rightarrow \frac{\mathrm{m}}{\mathrm{v}_{0}}=10 \mathrm{k}\)

\(\Rightarrow \mathrm{k}=\frac{\mathrm{m}}{10 \mathrm{v}_{0}}=\frac{10^{-2}}{10 \times 10}=10^{-4} \mathrm{kg} \mathrm{m}^{-1}\)