| Exam Board | AQA |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2009 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Finding constants from motion conditions |
| Difficulty | Standard +0.3 This is a standard M2 non-constant acceleration question with straightforward application of F=ma, followed by separating variables and integrating v^(-3/2). The algebra is routine and the question guides students through each step with clear targets to show. While it requires multiple techniques, these are all standard M2 procedures with no novel insight needed. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y)6.06a Variable force: dv/dt or v*dv/dx methods |
8 A stone, of mass $m$, is moving in a straight line along smooth horizontal ground.\\
At time $t$, the stone has speed $v$. As the stone moves, it experiences a total resistance force of magnitude $\lambda m v ^ { \frac { 3 } { 2 } }$, where $\lambda$ is a constant. No other horizontal force acts on the stone.
\begin{enumerate}[label=(\alph*)]
\item Show that
$$\frac { \mathrm { d } v } { \mathrm {~d} t } = - \lambda v ^ { \frac { 3 } { 2 } }$$
(2 marks)
\item The initial speed of the stone is $9 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Show that
$$v = \frac { 36 } { ( 2 + 3 \lambda t ) ^ { 2 } }$$
(7 marks)
\item Find, in terms of $\lambda$, the time taken for the speed of the stone to drop to $4 \mathrm {~ms} ^ { - 1 }$.
\end{enumerate}
\hfill \mbox{\textit{AQA M2 2009 Q8 [12]}}