| Exam Board | AQA |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2007 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable Force |
| Type | Air resistance kv - horizontal motion |
| Difficulty | Moderate -0.5 This is a straightforward application of Newton's second law followed by standard separation of variables to solve a first-order differential equation. Both parts are direct, single-step derivations with no problem-solving required—part (a) is immediate from F=ma, and part (b) is a textbook example of exponential decay that students will have seen multiple times in M2. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y)6.06a Variable force: dv/dt or v*dv/dx methods |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Part (a): Using \(F = ma\): \(-\lambda mv = ma = m\frac{dv}{dt}\); \(\therefore \frac{dv}{dt} = -\lambda v\) | M1, A1 | Condone no '−'; AG; Note: no use of \(m\) ⟹ no marks in (a) |
| Part (b): \(\int \frac{dv}{v} = -\lambda \int dt\); \(\ln v = -\lambda t + c\); \(v = Ce^{-\lambda t}\); When \(t = 0\), \(v = U \Rightarrow C = U\); \(v = Ue^{-\lambda t}\) | M1, A1, M1, A1 | Needs '+' c'; Needs correct working; AG |
| Total: 6 |
| Answer/Working | Marks | Guidance |
|---|---|---|
| **Part (a):** Using $F = ma$: $-\lambda mv = ma = m\frac{dv}{dt}$; $\therefore \frac{dv}{dt} = -\lambda v$ | M1, A1 | Condone no '−'; AG; Note: no use of $m$ ⟹ no marks in (a) |
| **Part (b):** $\int \frac{dv}{v} = -\lambda \int dt$; $\ln v = -\lambda t + c$; $v = Ce^{-\lambda t}$; When $t = 0$, $v = U \Rightarrow C = U$; $v = Ue^{-\lambda t}$ | M1, A1, M1, A1 | Needs '+' c'; Needs correct working; AG |
| | | **Total: 6** |7 A stone of mass $m$ is moving along the smooth horizontal floor of a tank which is filled with a viscous liquid. At time $t$, the stone has speed $v$. As the stone moves, it experiences a resistance force of magnitude $\lambda m v$, where $\lambda$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Show that
$$\frac { \mathrm { d } v } { \mathrm {~d} t } = - \lambda v$$
\item The initial speed of the stone is $U$.
Show that
$$v = U \mathrm { e } ^ { - \lambda t }$$
\end{enumerate}
\hfill \mbox{\textit{AQA M2 2007 Q7 [6]}}