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The Fundamental Equation of Dynamics

Problem 1.100

A motorboat of mass mm moves along a lake with velocity v0v_{0}. At the moment t=0t=0 the engine of the boat is shut down. Assuming the resistance of water to be proportional to the velocity of the boat F=−rv,\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=0t=0 ), during which its velocity decreases η\eta times.

Reveal Answer
1.100. (a) v=v0e−tr/m,t→∞v=v_{0} \mathrm{e}^{-t r / m}, \quad t \rightarrow \infty; (b) v=v0−sr/m,stotal =v=v_{0}-s r / m, \quad s_{\text {total }}= =mv0r;=\frac{m v_{0}}{r} ; \quad (c) ⟨v⟩=v0η−1η1ln⁡η\langle v\rangle=v_{0} \frac{\eta-1}{\eta_{1} \ln \eta}
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