| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Interception: verify/find meeting point (position vector method) |
| Difficulty | Moderate -0.3 This is a straightforward M1 kinematics question using position vectors. Part (a) requires basic application of r = r₀ + vt formula. Part (b) involves equating position vectors and solving simultaneous equations—standard procedure but requires careful algebra. Part (c) tests understanding of modeling assumptions. The question is slightly easier than average A-level because it's a routine application of vector kinematics with clear structure and the 'show that' in part (b) guides students to the answer. |
| Spec | 1.10d Vector operations: addition and scalar multiplication1.10e Position vectors: and displacement |
| Answer | Marks | Guidance |
|---|---|---|
| \((30 - 8t)i + (24t - 60)j\) m | A1 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| batsman is at \((4 \times 2.5)i = 10i\) | B1 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| ball will be affected by gravity ∴ not horizontal (may go over batsman) | B1 | B1 |
**(a)** $r = 4ti$ m
$s = (30i - 60j) + (8i + 24tj)$
$(30 - 8t)i + (24t - 60)j$ m | A1 | M1 | A1 |
**(b)** they will collide if coeffs. of $i$ and $j$ in $r$ and $s$ are equal
$4t = 30 - 8t$ and $24t - 60 = 0$
both are satisfied when $t = \frac{5}{2}$ so ball hits batsman
batsman is at $(4 \times 2.5)i = 10i$ | B1 | M1 | M1 A1 | A1 |
**(c)** ball travelling fast ∴ air resistance significant
ball will be affected by gravity ∴ not horizontal (may go over batsman) | B1 | B1 | (10) |
---During a cricket match, the batsman hits the ball and begins running with constant velocity $4\mathbf{i}$ m s$^{-1}$ to try and score a run. When the batsman is at the fixed origin $O$, the ball is thrown by a member of the opposing team with velocity $(^-8\mathbf{i} + 24\mathbf{j})$ m s$^{-1}$ from the point with position vector $(30\mathbf{i} - 60\mathbf{j})$ m, where $\mathbf{i}$ and $\mathbf{j}$ are horizontal perpendicular unit vectors. At time $t$ seconds after the ball is thrown, the position vectors of the batsman and the ball are $\mathbf{r}$ metres and $\mathbf{s}$ metres respectively.
In a model of the situation, the ball is assumed to travel horizontally and air resistance is considered to be negligible.
\begin{enumerate}[label=(\alph*)]
\item Find expressions for $\mathbf{r}$ and $\mathbf{s}$ in terms of $t$. [3 marks]
\item Show that the ball hits the batsman and find the position vector of the batsman when this occurs. [5 marks]
\item Write down two reasons why the assumptions used in these calculations are unlikely to provide a realistic model. [2 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 Q3 [10]}}