Question

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 $\overrightarrow{F}=-r\overrightarrow{v}$, find the velocity as a function of time and for how long the motorboat moved with the engine shutdown.

• A. $v=v_0{e}^{-\frac{tr}{m}},t\to \infty$
• B. $v=v_0{e}^{-\frac{tr}{2m}},t\to \infty$
• C. $v=v_0{e}^{-\frac{tr}{3m}},t\to \infty$
• D. $v=v_0{e}^{-\frac{2tr}{m}},t\to \infty$

Answer 1

Answer: (A)

$m\frac{d\overrightarrow{v}}{dt}=-r\overrightarrow{v}$ (from Newton's 2nd Law of motion)

$\frac{dv}{v}=-\frac{r}{m}dt$

Integrating, $lnv=-\frac{rt}{m}+lnc$, where $c$ is integrating constant.

$v=c{e}^{-\frac{rt}{m}}$

at $t=0,v=v_0,c=v_0$

$v=v_0{e}^{-\frac{rt}{m}}$

therefore velocity tends to zero when $t\to \infty$