A boat of mass m_1 moves on a lake at speed v_0 when its…
A boat of mass $m_1$ moves on a lake at speed $v_0$ when its engine is switched off. The water resistance is proportional to the speed, $F = -kv$. Find (1) how long the boat keeps moving, (2) its speed as a function of the distance travelled, (3) the total distance it covers.
Explanation of Method
A boat coasting against linear water resistance.
Step 1: Speed against time
\[ m_1\frac{dv}{dt} = -kv \;\Longrightarrow\; v(t) = v_0\,e^{-kt/m_1}. \]
\[\begin{gathered} v \to 0 \text{ only as } t \to \infty \\ \text{(in theory it never quite stops)} \end{gathered}\]
Step 2: Speed against distance
\[ m_1 v\frac{dv}{dx} = -kv \;\Longrightarrow\; \frac{dv}{dx} = -\frac{k}{m_1}. \]
\[v(x) = v_0 - \frac{k}{m_1}\,x\]
Step 3: Total distance
\[x_{\max} = \frac{m_1 v_0}{k}\]
Result
\[\boxed{\,\begin{gathered} v \to 0 \text{ only as } t \to \infty \\ \text{(in theory it never quite stops)} \end{gathered}\,}\]
\[\boxed{\,v(x) = v_0 - \frac{k}{m_1}\,x\,}\]
\[\boxed{\,x_{\max} = \frac{m_1 v_0}{k}\,}\]
Remark
Check. The original results are correct. The boat covers a finite distance in an infinite time: most of that distance is covered early, and the last part is crawled ever more slowly. (After $t = 3m_1/k$, for example, it has only 5% of its speed left.)