question_answer

The acceleration experienced by a moving boat after its engine is cut off, is given by\[\frac{dv}{dt}=\text{k}{{\upsilon }^{\text{3}}}\], where k is a constant. If \[{{\upsilon }_{0}}\] is the magnitude of the velocity at cut off, find the magnitude of the velocity at time t after the cut off.

Answer:

                Given, dv/dt = - kv3 or\[\text{d}\upsilon /{{\upsilon }^{\text{3}}}=-\text{kdt}\]. Integrating it within the condition of motion we have                 \[\int\limits_{{{v}_{0}}}^{v}{\frac{dv}{{{v}^{3}}}=-\int\limits_{0}^{t}{k\,\,dt}}\]  or            \[\left( -\frac{1}{2{{v}^{2}}} \right)_{{{v}_{0}}}^{v}=-k(t)_{0}^{t}\] or            \[\frac{1}{{{v}^{2}}}-\frac{1}{v_{0}^{2}}\]=2kt or v = \[\frac{{{v}_{0}}}{\sqrt{1+2kv_{0}^{2}t}}\]