| Exam Board | Edexcel |
|---|---|
| Module | Paper 3 (Paper 3) |
| Year | 2021 |
| Session | October |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (vectors) |
| Type | Find position by integrating velocity |
| Difficulty | Standard +0.3 This is a straightforward vector mechanics question requiring standard techniques: differentiation for acceleration, solving for direction condition, integration with initial conditions for position, and magnitude calculation. All steps are routine A-level Further Maths mechanics procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10h Vectors in kinematics: uniform acceleration in vector form3.02f Non-uniform acceleration: using differentiation and integration3.02g Two-dimensional variable acceleration |
| Answer | Marks | Guidance |
|---|---|---|
| Differentiate \(v\) wrt \(t\) | M1 | 3.1a |
| \(\frac{3}{2}t^{\frac{1}{2}} - 1 = \frac{3}{2}t^{\frac{1}{2}}\mathbf{i} - 2\mathbf{j}\) isw | A1 | 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{1}{3}t^2 = 2t\) | M1 | 2.1 |
| Solve for \(t\) | DM1 | 1.1b |
| \(t = \frac{9}{4}\) | A1 | 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| Integrate \(v\) wrt \(t\) | M1 | 3.1a |
| \(\mathbf{r} = 2t^{\frac{3}{2}}\mathbf{i} - t^2\mathbf{j} (+\mathbf{C})\) | A1 | 1.1b |
| \(t = 1\), \(\mathbf{r} = -\mathbf{j}\) \(\Rightarrow\) \(\mathbf{C} = -2\mathbf{i}\) so \(\mathbf{r} = 2t^{\frac{3}{2}}\mathbf{i} - t^2\mathbf{j} - 2\mathbf{i}\) | A1 | 2.2a |
| Answer | Marks | Guidance |
|---|---|---|
| \((3t^{\frac{1}{2}})^2 + (2t)^2 = 10^2\) or \((3t^{\frac{1}{2}})^2 + (2t)^2 = 100\) | M1 | 2.1 |
| \(9t + 4t^2 = 100\) | M(A)1 | 1.1b |
| \(t = 4\) | A1 | 1.1b |
| \(\mathbf{r} = 14\mathbf{i} - 16\mathbf{j}\) | M1 | 1.1b |
| \(\sqrt{14^2 + (-16)^2}\) | M1 | 3.1a |
| \(\sqrt{452}\) (\(2\sqrt{113}\)) (m) | A1 | 1.1b |
# Question 5
## 5(a)
Differentiate $v$ wrt $t$ | M1 | 3.1a
$\frac{3}{2}t^{\frac{1}{2}} - 1 = \frac{3}{2}t^{\frac{1}{2}}\mathbf{i} - 2\mathbf{j}$ isw | A1 | 1.1b
**Notes:** M1 — Both powers decreasing by 1 (M0 if vectors disappear but allow recovery). A1 — cao
(2)
## 5(b)
$\frac{1}{3}t^2 = 2t$ | M1 | 2.1
Solve for $t$ | DM1 | 1.1b
$t = \frac{9}{4}$ | A1 | 1.1b
**Notes:** M1 — Complete method, using $v$, to obtain an equation in $t$ only, allow a sign error. DM1 — Dependent on M1, solve for $t$. A1 — cao
(3)
## 5(c)
Integrate $v$ wrt $t$ | M1 | 3.1a
$\mathbf{r} = 2t^{\frac{3}{2}}\mathbf{i} - t^2\mathbf{j} (+\mathbf{C})$ | A1 | 1.1b
$t = 1$, $\mathbf{r} = -\mathbf{j}$ $\Rightarrow$ $\mathbf{C} = -2\mathbf{i}$ so $\mathbf{r} = 2t^{\frac{3}{2}}\mathbf{i} - t^2\mathbf{j} - 2\mathbf{i}$ | A1 | 2.2a
**Notes:** M1 — Both powers increasing by 1 (M0 if vectors disappear but allow recovery). A1 — Correct expression without $\mathbf{C}$. A1 — cao
(3)
## 5(d)
$(3t^{\frac{1}{2}})^2 + (2t)^2 = 10^2$ or $(3t^{\frac{1}{2}})^2 + (2t)^2 = 100$ | M1 | 2.1
$9t + 4t^2 = 100$ | M(A)1 | 1.1b
$t = 4$ | A1 | 1.1b
$\mathbf{r} = 14\mathbf{i} - 16\mathbf{j}$ | M1 | 1.1b
$\sqrt{14^2 + (-16)^2}$ | M1 | 3.1a
$\sqrt{452}$ ($2\sqrt{113}$) (m) | A1 | 1.1b
**Notes:** M1 — Use of Pythagoras on $v$ and 10 to set up equation in $t$. M(A)1 — Correct 3 term quadratic in $t$. A1 — cao. M1 — Substitute their numerical $t$ value into their $\mathbf{r}$. M1 — Use of Pythagoras to find the magnitude of their $\mathbf{r}$. A1 — cso
(6)
**Total: 14 marks**\begin{enumerate}
\item At time $t$ seconds, a particle $P$ has velocity $\mathbf { v } \mathrm { ms } ^ { - 1 }$, where
\end{enumerate}
$$\mathbf { v } = 3 t ^ { \frac { 1 } { 2 } } \mathbf { i } - 2 t \mathbf { j } \quad t > 0$$
(a) Find the acceleration of $P$ at time $t$ seconds, where $t > 0$\\
(b) Find the value of $t$ at the instant when $P$ is moving in the direction of $\mathbf { i } - \mathbf { j }$
At time $t$ seconds, where $t > 0$, the position vector of $P$, relative to a fixed origin $O$, is $\mathbf { r }$ metres.
When $t = 1 , \mathbf { r } = - \mathbf { j }$\\
(c) Find an expression for $\mathbf { r }$ in terms of $t$.\\
(d) Find the exact distance of $P$ from $O$ at the instant when $P$ is moving with speed $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
\hfill \mbox{\textit{Edexcel Paper 3 2021 Q5 [14]}}