| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Position from velocity and initial conditions |
| Difficulty | Moderate -0.3 This is a standard M1 kinematics question involving constant velocity vectors. It requires straightforward application of s = vt, magnitude calculations, and vector addition across multiple parts. While it has several steps (12 marks total), each individual calculation is routine with no conceptual challenges or novel problem-solving required—slightly easier than average due to its mechanical nature. |
| Spec | 1.10h Vectors in kinematics: uniform acceleration in vector form3.02a Kinematics language: position, displacement, velocity, acceleration |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(P\) has p.v. \(40i + 80j\), so \(OP = \sqrt{8000} = 89.4 \text{ m}\) | B1 M1 A1 | |
| (b) Speed from \(P\) to \(Q\) is \(3 \text{ m s}^{-1}\), so time \(= 25 \text{ s}\) | M1 A1 | |
| (c) \(OQ = 40i + 80j + 25(2 \cdot 4i - 1 \cdot 8j) = 100i + 35j\) | M1 A1 A1 | |
| (d) \(65(ai + bj) = 100i + 35j\) | \(a = \frac{20}{13}, b = \frac{7}{13}\) | M1 M1 A1 A1 |
(a) $P$ has p.v. $40i + 80j$, so $OP = \sqrt{8000} = 89.4 \text{ m}$ | B1 M1 A1
(b) Speed from $P$ to $Q$ is $3 \text{ m s}^{-1}$, so time $= 25 \text{ s}$ | M1 A1
(c) $OQ = 40i + 80j + 25(2 \cdot 4i - 1 \cdot 8j) = 100i + 35j$ | M1 A1 A1
(d) $65(ai + bj) = 100i + 35j$ | $a = \frac{20}{13}, b = \frac{7}{13}$ | M1 M1 A1 A1 | **12 marks**A boy starts at the corner $O$ of a rectangular playing field and runs across the field with constant velocity vector $(\mathbf{i} + 2\mathbf{j})$ ms$^{-1}$, where $\mathbf{i}$ and $\mathbf{j}$ are unit vectors in the directions of two perpendicular sides of the field. After 40 seconds, at the point $P$ in the field, he changes speed and direction so that his new velocity vector is $(2.4\mathbf{i} - 1.8\mathbf{j})$ ms$^{-1}$ and maintains this velocity until he reaches the point $Q$, where $PQ = 75$ m.
Calculate
\begin{enumerate}[label=(\alph*)]
\item the distance $OP$, [3 marks]
\item the time taken to travel from $P$ to $Q$, [2 marks]
\item the position vector of $Q$ relative to $O$. [3 marks]
\end{enumerate}
Another boy travels directly from $O$ to $Q$ with constant velocity $(a\mathbf{i} + b\mathbf{j})$ ms$^{-1}$, leaving $O$ and reaching $Q$ at the same times as the first boy.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Find the values of the constants $a$ and $b$. [4 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 Q4 [12]}}