| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Parallel or perpendicular vectors condition |
| Difficulty | Moderate -0.3 This is a straightforward M1 mechanics question requiring basic vector operations: (a) finding magnitude at t=0 is simple substitution, (b) parallel vectors condition requires equating components (standard technique), (c) is a simple interpretation question. All parts are routine applications of standard methods with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.10c Magnitude and direction: of vectors1.10h Vectors in kinematics: uniform acceleration in vector form |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(t = 0 \Rightarrow \mathbf{v} = 8\mathbf{i} + 6\mathbf{j}\) | M1 | |
| speed = \(\sqrt{8^2 + 6^2} = 10 \text{ ms}^{-1}\) | M1 A1 | |
| (b) parallel to \((\mathbf{i} + \mathbf{j})\) when \(3t^2 - 2t + 8 = 5t + 6\) | M1 | |
| i.e. \(3t^2 - 7t + 2 = 0\) | A1 | |
| \((3t - 1)(t - 2) = 0\) | M1 A1 | |
| \(t = \frac{1}{3}\) or 2 | A1 | |
| (c) e.g. improbably large values for the speed of the car | B1 | (9) |
(a) $t = 0 \Rightarrow \mathbf{v} = 8\mathbf{i} + 6\mathbf{j}$ | M1 |
speed = $\sqrt{8^2 + 6^2} = 10 \text{ ms}^{-1}$ | M1 A1 |
(b) parallel to $(\mathbf{i} + \mathbf{j})$ when $3t^2 - 2t + 8 = 5t + 6$ | M1 |
i.e. $3t^2 - 7t + 2 = 0$ | A1 |
$(3t - 1)(t - 2) = 0$ | M1 A1 |
$t = \frac{1}{3}$ or 2 | A1 |
(c) e.g. improbably large values for the speed of the car | B1 | (9) |3. In a simple model for the motion of a car, its velocity, $\mathbf { v }$, at time $t$ seconds, is given by
$$\mathbf { v } = \left( 3 t ^ { 2 } - 2 t + 8 \right) \mathbf { i } + ( 5 t + 6 ) \mathbf { j } \mathrm { ms } ^ { - 1 }$$
\begin{enumerate}[label=(\alph*)]
\item Calculate the speed of the car when $t = 0$.
\item Find the values of $t$ for which the velocity of the car is parallel to the vector $( \mathbf { i } + \mathbf { j } )$.
\item Why would this model not be appropriate for large values of $t$ ?
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 Q3 [9]}}