| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2023 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Total distance with direction changes |
| Difficulty | Standard +0.3 This is a straightforward kinematics question requiring standard techniques: substitution to verify rest, differentiation for acceleration, and integration with attention to direction changes. While part (c) requires identifying when velocity changes sign and computing two separate integrals, this is a routine skill at AS level with clear signposting in the question structure. |
| Spec | 3.02f Non-uniform acceleration: using differentiation and integration3.02g Two-dimensional variable acceleration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(15 - 3^2 - 2 \times 3 = 0\)* | B1* | AO 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Differentiate \(v\) wrt \(t\) | M1 | AO 2.1 |
| \(-2t - 2\) | A1 | AO 1.1b |
| \(8\ (\text{m s}^{-2})\) | A1 | AO 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Integrate \(v\) w.r.t. \(t\) | M1 | AO 1.1b |
| \(15t - \frac{1}{3}t^3 - t^2\) | A1 | AO 1.1b |
| Total distance \(= \left[15t - \frac{1}{3}t^3 - t^2\right]_0^3 - \left[15t - \frac{1}{3}t^3 - t^2\right]_3^4\) OR \(s_3 + (s_3 - s_4)\) | M1 | AO 3.1a |
| \(\frac{94}{3}\) (m) | A1 | AO 1.1b |
## Question 3:
### Part 3(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $15 - 3^2 - 2 \times 3 = 0$* | B1* | AO 1.1b |
Notes: Correct expression, correctly evaluated to give 0. OR $0 = 15 - t^2 - 2t$, $t = 3$
### Part 3(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Differentiate $v$ wrt $t$ | M1 | AO 2.1 |
| $-2t - 2$ | A1 | AO 1.1b |
| $8\ (\text{m s}^{-2})$ | A1 | AO 1.1b |
Notes: M1 for differentiating $v$ with at least two powers decreasing by 1. Final A1 must be positive. If 8 given as answer without working, can score all 3 marks.
### Part 3(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Integrate $v$ w.r.t. $t$ | M1 | AO 1.1b |
| $15t - \frac{1}{3}t^3 - t^2$ | A1 | AO 1.1b |
| Total distance $= \left[15t - \frac{1}{3}t^3 - t^2\right]_0^3 - \left[15t - \frac{1}{3}t^3 - t^2\right]_3^4$ **OR** $s_3 + (s_3 - s_4)$ | M1 | AO 3.1a |
| $\frac{94}{3}$ (m) | A1 | AO 1.1b |
Notes: M1 for integrating $v$ with at least two powers increasing by 1. Ignore $(+C)$. Accept 31(m) or better, must be positive. If indefinite integral $\left(15t - \frac{1}{3}t^3 - t^2\right)$ is never seen, score nothing even if correct answer appears (indicates calculator use).
---\begin{enumerate}
\item In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
\end{enumerate}
A fixed point $O$ lies on a straight line.\\
A particle $P$ moves along the straight line such that at time $t$ seconds, $t \geqslant 0$, after passing through $O$, the velocity of $P , v \mathrm {~ms} ^ { - 1 }$, is modelled as
$$v = 15 - t ^ { 2 } - 2 t$$
(a) Verify that $P$ comes to instantaneous rest when $t = 3$\\
(b) Find the magnitude of the acceleration of $P$ when $t = 3$\\
(c) Find the total distance travelled by $P$ in the interval $0 \leqslant t \leqslant 4$
\hfill \mbox{\textit{Edexcel AS Paper 2 2023 Q3 [8]}}