| Exam Board | Edexcel |
|---|---|
| Module | FM2 AS (Further Mechanics 2 AS) |
| Year | 2022 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Acceleration as function of velocity (separation of variables) |
| Difficulty | Standard +0.3 This is a straightforward Further Maths mechanics question requiring standard differentiation of an exponential function, substitution at t=0, and integration using v dv/dx = a. All techniques are routine for FM students, though the exponential and the v dv/dx method place it slightly above average A-level difficulty. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.07j Differentiate exponentials: e^(kx) and a^(kx)1.08d Evaluate definite integrals: between limits3.02a Kinematics language: position, displacement, velocity, acceleration3.02f Non-uniform acceleration: using differentiation and integration6.02b Calculate work: constant force, resolved component |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(a = \dfrac{dv}{dt} = \dfrac{1}{2} \times 6e^{2t}\) | M1 | Need to see evidence of attempt to differentiate \(v\) wrt \(t\), not just a statement of intent |
| \(= 2v + 1\) | A1* | Given answer correctly obtained |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(3 \ (\text{m s}^{-2})\) | B1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\dfrac{dx}{dt} = \dfrac{1}{2}(3e^{2t} - 1)\) and integrate | M1 | Set up differential equation and attempt to solve |
| \(x = \dfrac{1}{2}\!\left(\dfrac{3}{2}e^{2t} - t\right)(+C)\) | A1 | Condone missing \(C\) |
| Put either \(\dfrac{1}{2}(3e^{2t}-1) = 1\) or \(4\) and solve for \(t\) | M1 | Use at least one of the given speeds to find a \(t\) value |
| \(t = 0\) | A1 | cao |
| \(t = \dfrac{1}{2}\ln 3 \quad (0.549306\ldots)\) | A1 | \(0.55\) or better |
| Substitute their \(t\) values into their \(x\) expression and subtract | M1 | Need to see evidence of subtracting. M0 if using \(1\) and \(4\) |
| \(\dfrac{3}{2} - \dfrac{1}{4}\ln 3 \ \text{(m)}\) | A1 | cao |
## Question 4:
### Part 4(a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $a = \dfrac{dv}{dt} = \dfrac{1}{2} \times 6e^{2t}$ | M1 | Need to see evidence of attempt to differentiate $v$ wrt $t$, not just a statement of intent |
| $= 2v + 1$ | A1* | Given answer correctly obtained |
**Total: (2)**
---
### Part 4(b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $3 \ (\text{m s}^{-2})$ | B1 | cao |
**Total: (1)**
---
### Part 4(c):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\dfrac{dx}{dt} = \dfrac{1}{2}(3e^{2t} - 1)$ and integrate | M1 | Set up differential equation and attempt to solve |
| $x = \dfrac{1}{2}\!\left(\dfrac{3}{2}e^{2t} - t\right)(+C)$ | A1 | Condone missing $C$ |
| Put either $\dfrac{1}{2}(3e^{2t}-1) = 1$ or $4$ and solve for $t$ | M1 | Use at least one of the given speeds to find a $t$ value |
| $t = 0$ | A1 | cao |
| $t = \dfrac{1}{2}\ln 3 \quad (0.549306\ldots)$ | A1 | $0.55$ or better |
| Substitute their $t$ values into their $x$ expression and subtract | M1 | Need to see evidence of subtracting. M0 if using $1$ and $4$ |
| $\dfrac{3}{2} - \dfrac{1}{4}\ln 3 \ \text{(m)}$ | A1 | cao |
**Total: (7)**
---
**Overall Total: (10 marks)**\begin{enumerate}
\item A particle $P$ moves on the $x$-axis. At time $t$ seconds the velocity of $P$ is $v \mathrm {~ms} ^ { - 1 }$ in the direction of $x$ increasing, where
\end{enumerate}
$$v = \frac { 1 } { 2 } \left( 3 \mathrm { e } ^ { 2 t } - 1 \right) \quad t \geqslant 0$$
The acceleration of $P$ at time $t$ seconds is $a \mathrm {~ms} ^ { - 2 }$\\
(a) Show that $a = 2 v + 1$\\
(b) Find the acceleration of $P$ when $t = 0$\\
(c) Find the exact distance travelled by $P$ in accelerating from a speed of $1 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ to a speed of $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
\hfill \mbox{\textit{Edexcel FM2 AS 2022 Q4 [10]}}