Answer:
Here, \[\frac{dv}{dt}=-\text{k}{{\text{v}}^{\text{3}}}\] or \[\frac{dv}{{{v}^{3}}}=-\text{kdt}\]
Integrating it within the conditions of motion, we have
\[\int\limits_{{{v}_{0}}}^{v}{\frac{dv}{{{v}^{3}}}}=\int\limits_{0}^{t}{-k\,\,dt}\] or \[-\left(
\frac{1}{2{{v}^{2}}} \right)_{{{v}_{0}}}^{v}=-k\left( t \right)_{0}^{t}\] or \[\frac{1}{{{v}^{2}}}-\frac{1}{v_{0}^{2}}=2kt\]
or \[\frac{1}{{{v}^{2}}}=2kt+\frac{1}{v_{0}^{2}}=\frac{2kt\,\,v_{0}^{2}+1}{v_{0}^{2}}\] or \[v=\frac{{{v}_{0}}}{\sqrt{2kt\,\,v_{0}^{2}+1}}\]
You need to login to perform this action.
You will be redirected in
3 sec