| Exam Board | OCR |
|---|---|
| Module | H240/03 (Pure Mathematics and Mechanics) |
| Year | 2022 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Topic | Variable acceleration (vectors) |
| Type | Find force using F=ma |
| Difficulty | Standard +0.3 This is a standard A-level mechanics question involving vector differentiation and integration. Parts (a)-(c) require routine techniques: showing velocity is never zero, differentiating for acceleration, and using F=ma with parallel vectors. Part (d) involves integrating velocity to find position and calculating a bearing. While multi-step, each component uses familiar methods without requiring novel insight or complex problem-solving. |
| Spec | 1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication1.10h Vectors in kinematics: uniform acceleration in vector form3.02f Non-uniform acceleration: using differentiation and integration3.03d Newton's second law: 2D vectors |
In this question the unit vectors $\mathbf{i}$ and $\mathbf{j}$ are in the directions east and north respectively.
A particle $P$ is moving on a smooth horizontal surface under the action of a single force $\mathbf{F}$ N. At time $t$ seconds, where $t \geq 0$, the velocity $\mathbf{v} \mathrm{m s}^{-1}$ of $P$, relative to a fixed origin $O$, is given by
$$\mathbf{v} = (1 - 2t)\mathbf{i} + (2t^2 + t - 13)\mathbf{j}.$$
\begin{enumerate}[label=(\alph*)]
\item Show that $P$ is never stationary. [2]
\item Find, in terms of $\mathbf{i}$ and $\mathbf{j}$, the acceleration of $P$ at time $t$. [1]
\end{enumerate}
The mass of $P$ is 0.5 kg.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Determine the magnitude of $\mathbf{F}$ when $P$ is moving in the direction of the vector $-2\mathbf{i} + \mathbf{j}$. Give your answer correct to 3 significant figures. [5]
\end{enumerate}
When $t = 1$, $P$ is at the point with position vector $\frac{1}{6}\mathbf{j}$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Determine the bearing of $P$ from $O$ at time $t = 1.5$. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR H240/03 2022 Q12 [13]}}