| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2009 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Maximum or minimum velocity |
| Difficulty | Moderate -0.8 This is a straightforward mechanics question requiring basic calculus: differentiate to find maximum velocity (setting dv/dt = 0), and integrate to find displacement then solve for when x = 0. Both parts use standard A-level techniques with no conceptual challenges or multi-step reasoning. |
| Spec | 3.02a Kinematics language: position, displacement, velocity, acceleration3.02c Interpret kinematic graphs: gradient and area3.02f Non-uniform acceleration: using differentiation and integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{dv}{dt} = 8 - 2t\) | M1 | |
| \(8 - 2t = 0\) | M1 | |
| Max \(v = 8 \times 4 - 4^2 = 16 \text{ (ms}^{-1})\) | M1A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int 8t - t^2\, dt = 4t^2 - \frac{1}{3}t^3 (+c)\) | M1A1 | |
| \((t=0,\) displacement \(= 0 \Rightarrow c=0)\) | ||
| \(4T^2 - \frac{1}{3}T^3 = 0\) | DM1 | |
| \(T^2\left(4 - \frac{T}{3}\right) = 0 \Rightarrow T = 0, 12\) | DM1 | |
| \(T = 12\) (seconds) | A1 |
## Question 2:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{dv}{dt} = 8 - 2t$ | M1 | |
| $8 - 2t = 0$ | M1 | |
| Max $v = 8 \times 4 - 4^2 = 16 \text{ (ms}^{-1})$ | M1A1 | |
**Subtotal: (4)**
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int 8t - t^2\, dt = 4t^2 - \frac{1}{3}t^3 (+c)$ | M1A1 | |
| $(t=0,$ displacement $= 0 \Rightarrow c=0)$ | | |
| $4T^2 - \frac{1}{3}T^3 = 0$ | DM1 | |
| $T^2\left(4 - \frac{T}{3}\right) = 0 \Rightarrow T = 0, 12$ | DM1 | |
| $T = 12$ (seconds) | A1 | |
**Subtotal: (5) Total: [9]**
---2. At time $t = 0$ a particle $P$ leaves the origin $O$ and moves along the $x$-axis. At time $t$ seconds the velocity of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, where
$$v = 8 t - t ^ { 2 }$$
\begin{enumerate}[label=(\alph*)]
\item Find the maximum value of $v$.
\item Find the time taken for $P$ to return to $O$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 2009 Q2 [9]}}