| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2016 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (vectors) |
| Type | When moving parallel to given vector |
| Difficulty | Standard +0.3 This M2 question involves vector mechanics with polynomial functions but requires only routine techniques: equating components to find when velocity is in direction i+j, differentiating for acceleration, and integrating for displacement. The multi-part structure and vector notation add slight complexity, but each step follows standard procedures without requiring novel insight or difficult algebraic manipulation. |
| Spec | 3.02f Non-uniform acceleration: using differentiation and integration3.02g Two-dimensional variable acceleration |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Use \(\mathbf{v} = \lambda(\mathbf{i}+\mathbf{j})\): \(6T^2 + 6T = 3T^2 + 24\) | M1 | Form an equation in \(t\), \(T\) or \(\lambda\). \(\lambda^2 - 108\lambda + 2592 = 0\) |
| Solve for \(T\): \(3T^2 + 6T - 24 = 0\) | M1 | Simplify to quadratic in \(t\), \(T\) or \(\lambda\) and solve |
| \((T+4)(T-2) = 0\), \(T = 2\) | A1 | \(T = 2\) only |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Differentiate: \(\mathbf{a} = (12t + 6)\mathbf{i} + 6t\mathbf{j}\) | M1 | Majority of powers going down. Need to be considering both components |
| A1 | Correct in \(t\) or \(T\) | |
| \(= 30\mathbf{i} + 12\mathbf{j}\) (m s\(^{-2}\)) | A1 | CAO |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Integrate: \(\mathbf{r} = (2t^3 + 3t^2(+A))\mathbf{i} + (t^3 + 24t(+B))\mathbf{j}\) | M1 | Clear evidence of integration. Need to be considering both components. Do not need to see the constant(s) |
| A2 | \(-1\) each error | |
| \(\mathbf{OA} = 28\mathbf{i} + 56\mathbf{j}\) — Use their \(T\) | ||
| Distance \(= 28\sqrt{5} = 62.6\) (m) | DM1 | Dependent on previous M1. Use of Pythagoras on their \(\mathbf{OA}\) |
| A1 | 63 or better, \(\sqrt{3920}\) |
# Question 3:
## Part 3a:
| Working/Answer | Marks | Guidance |
|---|---|---|
| Use $\mathbf{v} = \lambda(\mathbf{i}+\mathbf{j})$: $6T^2 + 6T = 3T^2 + 24$ | M1 | Form an equation in $t$, $T$ or $\lambda$. $\lambda^2 - 108\lambda + 2592 = 0$ |
| Solve for $T$: $3T^2 + 6T - 24 = 0$ | M1 | Simplify to quadratic in $t$, $T$ or $\lambda$ and solve |
| $(T+4)(T-2) = 0$, $T = 2$ | A1 | $T = 2$ only |
## Part 3b:
| Working/Answer | Marks | Guidance |
|---|---|---|
| Differentiate: $\mathbf{a} = (12t + 6)\mathbf{i} + 6t\mathbf{j}$ | M1 | Majority of powers going down. Need to be considering both components |
| | A1 | Correct in $t$ or $T$ |
| $= 30\mathbf{i} + 12\mathbf{j}$ (m s$^{-2}$) | A1 | CAO |
## Part 3c:
| Working/Answer | Marks | Guidance |
|---|---|---|
| Integrate: $\mathbf{r} = (2t^3 + 3t^2(+A))\mathbf{i} + (t^3 + 24t(+B))\mathbf{j}$ | M1 | Clear evidence of integration. Need to be considering both components. Do not need to see the constant(s) |
| | A2 | $-1$ each error |
| $\mathbf{OA} = 28\mathbf{i} + 56\mathbf{j}$ — Use their $T$ | | |
| Distance $= 28\sqrt{5} = 62.6$ (m) | DM1 | Dependent on previous M1. Use of Pythagoras on their $\mathbf{OA}$ |
| | A1 | 63 or better, $\sqrt{3920}$ |
---3.At time $t$ seconds( $t \geqslant 0$ )a particle $P$ has velocity $\mathbf { v } \mathrm { ms } ^ { - 1 }$ ,where
When $t = 0$ the particle $P$ is at the origin $O$ .At time $T$ seconds,$P$ is at the point $A$ and $\mathbf { v } = \lambda ( \mathbf { i } + \mathbf { j } )$ ,where $\lambda$ is a constant.
Find
\begin{enumerate}[label=(\alph*)]
\item the value of $T$ ,
\item the acceleration of $P$ as it passes through the point $A$ ,
\item the distance $O A$ .
$$\mathbf { v } = \left( 6 t ^ { 2 } + 6 t \right) \mathbf { i } + \left( 3 t ^ { 2 } + 24 \right) \mathbf { j }$$
的 When $t = 0$ the particle $P$ is at the origin $O$ .At time $T$ seconds,$P$ is at the point $A$ and\\
$\mathbf { v } = \lambda ( \mathbf { i } + \mathbf { j } )$ ,where $\lambda$ is a constant. Find\\
(a)the value of $T$ ,\\
$\_\_\_\_$ "
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 2016 Q3 [11]}}