| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Newton's second law with vector forces (find acceleration or force) |
| Difficulty | Moderate -0.8 This is a straightforward M1 mechanics question requiring basic vector manipulation and Newton's second law. Part (a) uses Pythagoras to find λ from the given speed, and part (b) applies F=ma with constant acceleration to find the force vector. Both parts involve routine calculations with no problem-solving insight required, making it easier than average A-level questions. |
| Spec | 1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication3.03d Newton's second law: 2D vectors3.03f Weight: W=mg |
| Answer | Marks |
|---|---|
| Speed = 17 = mag. of vel. = \(\sqrt{8^2 + \lambda^2}\) | M1 |
| \(\lambda^2 = 289 - 64 = 225; \lambda > 0\) so \(\lambda = 15\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(a = \frac{\Delta v}{t} = \frac{1}{2}[(3\mathbf{i} + 5\mathbf{j}) - (8\mathbf{i} + 15\mathbf{j})] = \mathbf{i} - 2\mathbf{j}\) | M2 A1 | |
| \(\mathbf{F} = m\mathbf{a} = 2(\mathbf{i} - 2\mathbf{j}) = 2\mathbf{i} + 2\mathbf{j}\) so \(\mu = -2\) | M1 A1 | (8) |
**Part (a)**
Speed = 17 = mag. of vel. = $\sqrt{8^2 + \lambda^2}$ | M1 |
$\lambda^2 = 289 - 64 = 225; \lambda > 0$ so $\lambda = 15$ | M1 A1 |
**Part (b)**
$a = \frac{\Delta v}{t} = \frac{1}{2}[(3\mathbf{i} + 5\mathbf{j}) - (8\mathbf{i} + 15\mathbf{j})] = \mathbf{i} - 2\mathbf{j}$ | M2 A1 |
$\mathbf{F} = m\mathbf{a} = 2(\mathbf{i} - 2\mathbf{j}) = 2\mathbf{i} + 2\mathbf{j}$ so $\mu = -2$ | M1 A1 | (8)\begin{enumerate}
\item At time $t = 0$, a particle of mass 2 kg has velocity $( 8 \mathbf { i } + \lambda \mathbf { j } ) \mathrm { ms } ^ { - 1 }$ where $\mathbf { i }$ and $\mathbf { j }$ are horizontal perpendicular unit vectors and $\lambda > 0$.
\end{enumerate}
Given that the speed of the particle at time $t = 0$ is $17 \mathrm {~m} \mathrm {~s} ^ { - 1 }$,\\
(a) find the value of $\lambda$.
The particle experiences a constant retarding force $\mathbf { F }$ so that when $t = 5$, it has velocity $( 3 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$.\\
(b) Show that $\mathbf { F }$ can be written in the form $\mu ( \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }$ where $\mu$ is a constant which you should find.\\
(5 marks)\\
\hfill \mbox{\textit{Edexcel M1 Q1 [8]}}