| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2003 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Variable acceleration with initial conditions |
| Difficulty | Standard +0.3 This is a standard M2 integration question requiring integration of acceleration to find velocity, then solving a quadratic to find rest points, and finally integrating velocity to find displacement. While it involves multiple steps (11 marks total), each step follows routine procedures with no novel insight required—slightly easier than average due to straightforward setup and clean arithmetic. |
| Spec | 1.08d Evaluate definite integrals: between limits3.02a Kinematics language: position, displacement, velocity, acceleration3.02f Non-uniform acceleration: using differentiation and integration |
| Answer | Marks | Guidance |
|---|---|---|
| \(v = \int a \, dt = 2t^2 - 8t \,(+c)\) | M1 A1 | |
| Using \(v = 6\), \(t = 0\): \(v = 2t^2 - 8t + 6\) | M1 A1 | (4 marks) |
| \(v = 0 \Rightarrow 2t^2 - 8t + 6 = 0, \Rightarrow t = 1, 3\) | M1 A1 | |
| \(S = \int (2t^2 - 8t + 6) \, dt = \left[\frac{2}{3}t^3 - 4t^2 + 6t\right]\) | M1 A2, 1, 0 | |
| \(= 0 - 2\frac{2}{3}\) | M1 | |
| Distance is \((\pm)2\frac{2}{3}\) m | A1 | (7 marks) |
## Part (a)
$v = \int a \, dt = 2t^2 - 8t \,(+c)$ | M1 A1 |
Using $v = 6$, $t = 0$: $v = 2t^2 - 8t + 6$ | M1 A1 | **(4 marks)**
$v = 0 \Rightarrow 2t^2 - 8t + 6 = 0, \Rightarrow t = 1, 3$ | M1 A1 |
$S = \int (2t^2 - 8t + 6) \, dt = \left[\frac{2}{3}t^3 - 4t^2 + 6t\right]$ | M1 A2, 1, 0 |
$= 0 - 2\frac{2}{3}$ | M1 |
Distance is $(\pm)2\frac{2}{3}$ m | A1 | **(7 marks)**
**(11 marks total)**
---A particle $P$ moves on the $x$-axis. The acceleration of $P$ at time $t$ seconds is $(4t - 8)$ m s$^{-2}$, measured in the direction of $x$ increasing. The velocity of $P$ at time $t$ seconds is $v$ m s$^{-1}$. Given that $v = 6$ when $t = 0$, find
\begin{enumerate}[label=(\alph*)]
\item $v$ in terms of $t$, [4]
\item the distance between the two points where $P$ is instantaneously at rest. [7]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 2003 Q5 [11]}}