Problem source

A skater of mass $M$ is skating inattentively on a smooth frozen canal. She suddenly realises that she is heading perpendicularly towards the straight canal bank at speed $V$. She is at a distance $d$ from the bank and can choose one of two methods of trying to avoid it; either she can apply a force of constant magnitude $F$, acting at right-angles to her velocity, so that she travels in a circle; or she can apply a force of magnitude $\frac{1}{2}F(V^{2}+v^{2})/V^{2}$ directly backwards, where $v$ is her instantaneous speed. Treating the skater as a particle, find the set of values of $d$ for which she can avoid hitting the bank. Comment \textbf{briefly} on the assumption that the skater is a particle.

Solution source

Suppose she applies a force of magnitude $\frac{1}{2}F(V^{2}+v^{2})/V^{2}$ backwards,

then
\begin{align*}
&& M v \frac{dv}{dx} &= -\frac{1}{2}F(V^{2}+v^{2})/V^{2} \\
\Rightarrow && M\int_{V}^0 \frac{2v}{V^2+ v^2} \d v &= - \frac{F}{V^2} x \\
\Rightarrow && M \left [ -\log(V^2+v^2) \right]_0^V &= -\frac{Fx}{V^2} \\
\Rightarrow && -M \ln 2&= -\frac{Fx}{V^2}
\end{align*}
Therefore she will stop quickly enough if $d > \frac{V^2M \ln 2}{F}$

If she attempts to use the right-angled method, then she will travel a distance at most $r$ where $r$ is the radius of her circle:

\begin{align*}
&& F &= M \frac{V^2}{r} \\
\Rightarrow && r &= \frac{MV^2}{F}
\end{align*}

Therefore she can always avoid the wall if $d > \frac{MV^2}{F}$.

There are two potential issues with being a particle.

Firstly we would need to account for any variation in the distance to the wall (which could be accounted for by changing $d$). Secondly when she enters circular motion she will rotate and therefore we might need to consider her inertia as well as just her velocity when modelling.