| Exam Board | AQA |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2014 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable Force |
| Type | Air resistance with other powers |
| Difficulty | Standard +0.8 This M2 question requires setting up and solving a differential equation with variable resistance (F = 0.3mv³), separating variables with a non-standard integrand (1/v³), and then integrating v(t) to find distance. While the calculus techniques are A-level standard, the cubic resistance law makes this less routine than typical linear resistance problems, and the multi-step nature with 13 total marks indicates above-average difficulty for M2. |
| Spec | 1.08h Integration by substitution6.06a Variable force: dv/dt or v*dv/dx methods |
A puck, of mass $m$ kg, is moving in a straight line across smooth horizontal ice. At time $t$ seconds, the puck has speed $v \text{ m s}^{-1}$. As the puck moves, it experiences an air resistance force of magnitude $0.3mv^3$ newtons, until it comes to rest. No other horizontal forces act on the puck.
When $t = 0$, the speed of the puck is $8 \text{ m s}^{-1}$.
Model the puck as a particle.
\begin{enumerate}[label=(\alph*)]
\item Show that
$$v = (4 - 0.2t)^{\frac{3}{2}}$$
[6 marks]
\item Find the value of $t$ when the puck comes to rest. [2 marks]
\item Find the distance travelled by the puck as its speed decreases from $8 \text{ m s}^{-1}$ to zero. [5 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA M2 2014 Q6 [13]}}