| Exam Board | Edexcel |
|---|---|
| Module | FM2 AS (Further Mechanics 2 AS) |
| Year | 2021 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Displacement from velocity by integration |
| Difficulty | Standard +0.8 This is a Further Maths mechanics question requiring integration of trigonometric velocity to find displacement, analysis of motion bounds, and calculating time intervals. While the integration itself is standard (∫sin 2t dt), parts (b) and (c) require understanding of periodic motion, finding extrema, and careful analysis of when x > 3 over multiple periods. The multi-step reasoning and periodic motion analysis elevate this above routine A-level questions but it remains a structured textbook-style problem. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.08a Fundamental theorem of calculus: integration as reverse of differentiation1.08d Evaluate definite integrals: between limits3.02a Kinematics language: position, displacement, velocity, acceleration3.02b Kinematic graphs: displacement-time and velocity-time3.02c Interpret kinematic graphs: gradient and area3.02f Non-uniform acceleration: using differentiation and integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Instantaneous rest: \(v = 5\sin 2t = 0\) | M1 | Solve \(v = 0\) to find the first value of \(t > 0\) |
| \(\Rightarrow t = \frac{\pi}{2}\) | A1 | Or equivalent (accept \(2t = \pi\)) |
| Differentiate to obtain \(a\): \(a = 10\cos 2t\ (= 10\cos\pi = -10)\) | M1 | Use \(a = \frac{\mathrm{d}v}{\mathrm{d}t}\) and substitute for \(t\) |
| \(10\ (\text{ms}^{-2})\) towards \(O\) | A1 | Correct only |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Integrate to obtain \(x\): \(x = -\frac{5}{2}\cos 2t\ (+C)\) | M1 | Use \(x = \int v\,\mathrm{d}t\). Condone missing \(+C\) |
| Use boundary conditions: \(t = 0,\ x = 1 \Rightarrow C = \frac{7}{2}\) | M1 | Use boundary conditions to find \(C\) |
| \(x = \frac{7}{2} - \frac{5}{2}\cos 2t\) | A1 | Any equivalent form |
| \(-1 \leq \cos 2t \leq 1 \Rightarrow 1 \leq x \leq 6\) | A1* | Deduce given answer from correct working |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x = \frac{7}{2} - \frac{5}{2}\cos 2t = 3 \Rightarrow \cos 2t = \frac{1}{5},\ t = 0.6847...\) | M1 | Use trig to find a relevant value of \(t\) |
| Total time \(= 4\pi - 8t\) | M1 | Correct method for the total time |
| \(= 7.1\ \text{(s)}\) or better | A1 | \(7.1\ \text{(s)}\) or better \(7.08861...\) |
## Question 4(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Instantaneous rest: $v = 5\sin 2t = 0$ | M1 | Solve $v = 0$ to find the first value of $t > 0$ |
| $\Rightarrow t = \frac{\pi}{2}$ | A1 | Or equivalent (accept $2t = \pi$) |
| Differentiate to obtain $a$: $a = 10\cos 2t\ (= 10\cos\pi = -10)$ | M1 | Use $a = \frac{\mathrm{d}v}{\mathrm{d}t}$ and substitute for $t$ |
| $10\ (\text{ms}^{-2})$ towards $O$ | A1 | Correct only |
## Question 4(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Integrate to obtain $x$: $x = -\frac{5}{2}\cos 2t\ (+C)$ | M1 | Use $x = \int v\,\mathrm{d}t$. Condone missing $+C$ |
| Use boundary conditions: $t = 0,\ x = 1 \Rightarrow C = \frac{7}{2}$ | M1 | Use boundary conditions to find $C$ |
| $x = \frac{7}{2} - \frac{5}{2}\cos 2t$ | A1 | Any equivalent form |
| $-1 \leq \cos 2t \leq 1 \Rightarrow 1 \leq x \leq 6$ | A1* | Deduce given answer from correct working |
## Question 4(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x = \frac{7}{2} - \frac{5}{2}\cos 2t = 3 \Rightarrow \cos 2t = \frac{1}{5},\ t = 0.6847...$ | M1 | Use trig to find a relevant value of $t$ |
| Total time $= 4\pi - 8t$ | M1 | Correct method for the total time |
| $= 7.1\ \text{(s)}$ or better | A1 | $7.1\ \text{(s)}$ or better $7.08861...$ |\begin{enumerate}
\item A particle $P$ moves on the $x$-axis. At time $t$ seconds, $t \geqslant 0 , P$ is $x$ metres from the origin $O$ and moving with velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in the direction of $x$ increasing, where
\end{enumerate}
$$v = 5 \sin 2 t$$
When $t = 0 , x = 1$ and $P$ is at rest.\\
(a) Find the magnitude and direction of the acceleration of $P$ at the instant when $P$ is next at rest.\\
(b) Show that $1 \leqslant x \leqslant 6$\\
(c) Find the total time, in the first $4 \pi$ seconds of the motion, for which $P$ is more than 3 metres from $O$
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\hfill \mbox{\textit{Edexcel FM2 AS 2021 Q4 [11]}}