| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2011 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Variable acceleration with initial conditions |
| Difficulty | Moderate -0.3 This is a standard M2 variable acceleration question requiring straightforward integration twice with given initial conditions. While it involves polynomial integration and solving a quartic equation, these are routine techniques for M2 students with no novel problem-solving required—slightly easier than average A-level difficulty. |
| Spec | 3.02f Non-uniform acceleration: using differentiation and integration3.02g Two-dimensional variable acceleration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(a = 4t^3 - 12t\) | ||
| Convincing attempt to integrate | M1 | |
| \(v = t^4 - 6t^2 (+c)\) | A1 | |
| Use initial condition to get \(v = t^4 - 6t^2 + 8 \text{ ms}^{-1}\) | A1 | |
| (3) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Convincing attempt to integrate | M1 | |
| \(s = \frac{t^5}{5} - 2t^3 + 8t\ (+0)\) | A1ft | Integral of their \(v\) |
| (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Set their \(v = 0\) | M1 | |
| Solve a quadratic in \(t^2\) | DM1 | |
| \((t^2-2)(t^2-4)=0 \Rightarrow\) at rest when \(t=\sqrt{2},\ t=2\) | A1 | |
| (3) [8] |
## Question 3:
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $a = 4t^3 - 12t$ | | |
| Convincing attempt to integrate | M1 | |
| $v = t^4 - 6t^2 (+c)$ | A1 | |
| Use initial condition to get $v = t^4 - 6t^2 + 8 \text{ ms}^{-1}$ | A1 | |
| | **(3)** | |
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Convincing attempt to integrate | M1 | |
| $s = \frac{t^5}{5} - 2t^3 + 8t\ (+0)$ | A1ft | Integral of their $v$ |
| | **(2)** | |
### Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Set their $v = 0$ | M1 | |
| Solve a quadratic in $t^2$ | DM1 | |
| $(t^2-2)(t^2-4)=0 \Rightarrow$ at rest when $t=\sqrt{2},\ t=2$ | A1 | |
| | **(3) [8]** | |
---3. A particle moves along the $x$-axis. At time $t = 0$ the particle passes through the origin with speed $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in the positive $x$-direction. The acceleration of the particle at time $t$ seconds, $t \geqslant 0$, is $\left( 4 t ^ { 3 } - 12 t \right) \mathrm { m } \mathrm { s } ^ { - 2 }$ in the positive $x$-direction.
Find
\begin{enumerate}[label=(\alph*)]
\item the velocity of the particle at time $t$ seconds,
\item the displacement of the particle from the origin at time $t$ seconds,
\item the values of $t$ at which the particle is instantaneously at rest.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 2011 Q3 [8]}}