| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Marks | 11 |
| 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 standard M1 relative velocity question with structured parts guiding students through position vectors, relative position, and collision conditions. The 'show that' format reduces difficulty, and the method is straightforward: equate components to zero and solve simultaneous equations. Slightly easier than average due to scaffolding and routine application of kinematics formulas. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors3.02e Two-dimensional constant acceleration: with vectors |
| Answer | Marks | Guidance |
|---|---|---|
| (a) initially, \(A\) is at \(6i\) and travels \((-4T\mathbf{i} + T\mathbf{j})\) km in \(T\) hours. \(T\) hours after midday, \(A\) is at \((6 - 4T)\mathbf{i} + T\mathbf{j}\) km. Initially \(B\) is at \(3\mathbf{j}\) and travels \((4T\mathbf{i} - 3T\mathbf{j})\) km in \(T\) hours. \(T\) hours after midday, \(B\) is at \(4T\mathbf{i} + (3 - 3T)\mathbf{j}\) km | M2, A1, M2, A1 | |
| (b) pos\(^n\) \(B\) rel. to \(A\) is \([4T - (6-4T)]\mathbf{i} + [(3-3T) - T]\mathbf{j}\) i.e. \([(8T-6)\mathbf{i} + (3-4T)\mathbf{j}]\) km | M1, A1 | |
| (c) they will collide if coeffs. of \(\mathbf{i}\) and \(\mathbf{j}\) in part (b) are both zero \(8T - 6 = 0\) and \(3 - 4T = 0\) are both satisfied when \(T = \frac{3}{4}\) i.e. collision at 12:45 p.m. | M1, A1, A1 | (11) |
| (a) initially, $A$ is at $6i$ and travels $(-4T\mathbf{i} + T\mathbf{j})$ km in $T$ hours. $T$ hours after midday, $A$ is at $(6 - 4T)\mathbf{i} + T\mathbf{j}$ km. Initially $B$ is at $3\mathbf{j}$ and travels $(4T\mathbf{i} - 3T\mathbf{j})$ km in $T$ hours. $T$ hours after midday, $B$ is at $4T\mathbf{i} + (3 - 3T)\mathbf{j}$ km | M2, A1, M2, A1 | |
| (b) pos$^n$ $B$ rel. to $A$ is $[4T - (6-4T)]\mathbf{i} + [(3-3T) - T]\mathbf{j}$ i.e. $[(8T-6)\mathbf{i} + (3-4T)\mathbf{j}]$ km | M1, A1 | |
| (c) they will collide if coeffs. of $\mathbf{i}$ and $\mathbf{j}$ in part (b) are both zero $8T - 6 = 0$ and $3 - 4T = 0$ are both satisfied when $T = \frac{3}{4}$ i.e. collision at 12:45 p.m. | M1, A1, A1 | (11) |5. The unit vectors $\mathbf { i }$ and $\mathbf { j }$ are due east and due north respectively.
At midday a motor boat $A$ is 6 km east of a fixed origin $O$ and is moving with constant velocity ( ${ } ^ { - } 4 \mathbf { i } + \mathbf { j }$ ) $\mathrm { km } \mathrm { h } ^ { - 1 }$. At the same time, another boat $B$ is 3 km north of $O$ and is moving with uniform velocity $( 4 \mathbf { i } - 3 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$.
\begin{enumerate}[label=(\alph*)]
\item Show that, at time $T$ hours after midday, the position vector of $A$ is $[ ( 6 - 4 T ) \mathbf { i } + T \mathbf { j } ] \mathrm { km }$ and find a similar expression for the position vector of $B$ at this time.
\item Hence show that, at time $T$, the position vector of $B$ relative to $A$ is
$$[ ( 8 T - 6 ) \mathbf { i } + ( 3 - 4 T ) \mathbf { j } ] \mathrm { km }$$
\item By using your answer to part (b), or otherwise, show that the boats would collide if they continued at the same velocities and find the time at which the collision would occur.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 Q5 [11]}}