| Exam Board | OCR MEI |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2005 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (vectors) |
| Type | Find force using F=ma |
| Difficulty | Moderate -0.8 This is a straightforward mechanics question requiring only routine differentiation of a position vector twice to find acceleration, then direct application of F=ma. Both parts are standard textbook exercises with no problem-solving or insight required—simpler than average A-level questions. |
| Spec | 3.02g Two-dimensional variable acceleration3.03d Newton's second law: 2D vectors |
| Answer | Marks | Guidance |
|---|---|---|
| Differentiate \(v = 2ti + (5 - 4t)j\) | M1 | At least 1 cpt correct |
| A1 | Award for RHS seen | |
| Differentiate \(a = 2i - 4j\) | M1 | Do not award if \(i\) and \(j\) lost in \(v\). At least 1 cpt correct. FT FT from their 2 component \(v\) |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\mathbf{F} + 12j = 4(2i - 4j)\) | M1 | N2L. Allow \(F = mg a\). No extra forces. Allow 12j omitted |
| A1 | Allow wrong signs otherwise correct with their vector \(a\) | |
| \(\mathbf{F} = 8i - 28j\) | A1 | cao |
| Total: 3 |
**Part (i)**
| Differentiate $v = 2ti + (5 - 4t)j$ | M1 | At least 1 cpt correct |
|---|---|---|
| | A1 | Award for RHS seen |
| Differentiate $a = 2i - 4j$ | M1 | Do not award if $i$ and $j$ lost in $v$. At least 1 cpt correct. FT FT from their 2 component $v$ |
| | | **Total: 4** |
**Part (ii)**
| $\mathbf{F} + 12j = 4(2i - 4j)$ | M1 | N2L. Allow $F = mg a$. No extra forces. Allow 12j omitted |
| | A1 | Allow wrong signs otherwise correct with their vector $a$ |
| $\mathbf{F} = 8i - 28j$ | A1 | cao |
| | | **Total: 3** |
---1 The position vector, $\mathbf { r }$, of a particle of mass 4 kg at time $t$ is given by
$$\mathbf { r } = t ^ { 2 } \mathbf { i } + \left( 5 t - 2 t ^ { 2 } \right) \mathbf { j } ,$$
where $\mathbf { i }$ and $\mathbf { j }$ are the standard unit vectors, lengths are in metres and time is in seconds.\\
(i) Find an expression for the acceleration of the particle.
The particle is subject to a force $\mathbf { F }$ and a force $12 \mathbf { j } \mathbf { N }$.\\
(ii) Find $\mathbf { F }$.
\hfill \mbox{\textit{OCR MEI M1 2005 Q1 [7]}}