| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Total distance with direction changes |
| Difficulty | Moderate -0.3 This is a straightforward M2 kinematics question requiring basic calculus operations: substituting t=0 for initial velocity, solving a quadratic equation to find when v=0, and integrating velocity to find displacement. All techniques are standard with no conceptual challenges or novel problem-solving required, making it slightly easier than average. |
| Spec | 3.02f Non-uniform acceleration: using differentiation and integration |
| Answer | Marks |
|---|---|
| when \(t = 0\), \(v = 4 \text{ ms}^{-1}\) | A1 |
| Answer | Marks |
|---|---|
| particle at rest when \(2t^2 - 9t + 4 = 0\) i.e. \((2t-1)(t-4) = 0\) | M1 A1 |
| \(t = \frac{1}{2}, 4\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(s = \int v dt = \frac{2}{3}t^3 - \frac{9}{2}t^2 + 4t + c\) | M1 A1 | |
| when \(t = 0\), \(s = 9\) so \(c = 9\) \(\therefore s = \frac{2}{3}t^3 - \frac{9}{2}t^2 + 4t + 9\) | A1 | |
| disp. when \(t = 6\) is \(\frac{2}{3}(6)^3 - \frac{9}{2}(6)^2 + 4(6) + 9\) | M1 | |
| \(= 144 - 162 + 24 + 9 = 15 \text{ m}\) | A1 | (9) |
**Part (a):**
when $t = 0$, $v = 4 \text{ ms}^{-1}$ | A1 |
**Part (b):**
particle at rest when $2t^2 - 9t + 4 = 0$ i.e. $(2t-1)(t-4) = 0$ | M1 A1 |
$t = \frac{1}{2}, 4$ | A1 |
**Part (c):**
$s = \int v dt = \frac{2}{3}t^3 - \frac{9}{2}t^2 + 4t + c$ | M1 A1 |
when $t = 0$, $s = 9$ so $c = 9$ $\therefore s = \frac{2}{3}t^3 - \frac{9}{2}t^2 + 4t + 9$ | A1 |
disp. when $t = 6$ is $\frac{2}{3}(6)^3 - \frac{9}{2}(6)^2 + 4(6) + 9$ | M1 |
$= 144 - 162 + 24 + 9 = 15 \text{ m}$ | A1 | (9)
---3. A particle moves in a straight horizontal line such that its velocity, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, at time $t$ seconds is given by $v = 2 t ^ { 2 } - 9 t + 4$. Initially, the particle has displacement 9 m from a fixed point $O$ on the line.
\begin{enumerate}[label=(\alph*)]
\item Find the initial velocity of the particle.
\item Show that the particle is at rest when $t = 4$ and find the other value of $t$ when it is at rest.
\item Find the displacement of the particle from $O$ when $t = 6$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 Q3 [9]}}