All Problems The Fundamental Equation of Dynamics
Problem 1.100
A motorboat of mass \(m\) moves along a lake with velocity \(v_{0}\). At the moment \(t=0\) the engine of the boat is shut down. Assuming the resistance of water to be proportional to the velocity of the boat \(\mathbf{F}=-r \mathbf{v},\) find (a) how long the motorboat moved with the shutdown engine; (b) the velocity of the motorboat as a function of the distance covered with the shutdown engine, as well as the total distance covered till the complete stop; (c) the mean velocity of the motorboat over the time interval (beginning with the moment \(t=0\) ), during which its velocity decreases \(\eta\) times.
1.100. (a) \(v=v_{0} \mathrm{e}^{-t r / m}, \quad t \rightarrow \infty\); (b) \(v=v_{0}-s r / m, \quad s_{\text {total }}=\) \(=\frac{m v_{0}}{r} ; \quad\) (c) \(\langle v\rangle=v_{0} \frac{\eta-1}{\eta_{1} \ln \eta}\)