| Exam Board | AQA |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2006 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions |
| Type | Collision with vector velocities |
| Difficulty | Moderate -0.3 This is a straightforward momentum conservation problem with vector notation. Part (a) requires simple vector addition of momenta (mass × velocity). Part (b) applies conservation of momentum in two dimensions to find m and k. While it involves vectors and two components, the steps are routine and mechanical—no geometric insight or novel problem-solving is needed, making it slightly easier than average. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution6.03b Conservation of momentum: 1D two particles |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(m(5\mathbf{i} - 3\mathbf{j}) + 0.2(2\mathbf{i} + 3\mathbf{j})\) | M1 | Momentum terms added |
| A1 | All correct | Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \((0.2 + m)(k\mathbf{i} + \mathbf{j})\) | B1 | Seen or used to find \(m\) |
| Use of conservation of momentum | M1 | Used with candidate's expressions in 2D equation or used to give one of the 1D equations below |
| \(-3m + 0.6 = 0.2 + m\) | ||
| \(m = 0.1\) | A1 | Full verification accepted, CAO |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(5m + 0.4 = 0.2k + mk\) | A1 | |
| substitute \(m\) | m1 | |
| \(k = 3\) | A1 |
## Question 8:
### Part (a)
| Working | Marks | Guidance |
|---------|-------|----------|
| $m(5\mathbf{i} - 3\mathbf{j}) + 0.2(2\mathbf{i} + 3\mathbf{j})$ | M1 | Momentum terms added |
| | A1 | All correct | **Total: 2** |
### Part (b)(i)
| Working | Marks | Guidance |
|---------|-------|----------|
| $(0.2 + m)(k\mathbf{i} + \mathbf{j})$ | B1 | Seen or used to find $m$ |
| Use of conservation of momentum | M1 | Used with candidate's expressions in 2D equation or used to give one of the 1D equations below |
| $-3m + 0.6 = 0.2 + m$ | | |
| $m = 0.1$ | A1 | Full verification accepted, CAO | **Total: 3** |
### Part (b)(ii)
| Working | Marks | Guidance |
|---------|-------|----------|
| $5m + 0.4 = 0.2k + mk$ | A1 | |
| substitute $m$ | m1 | |
| $k = 3$ | A1 | | **Total: 3** |8 Two particles, $A$ and $B$, are moving on a smooth horizontal surface.\\
The particle $A$ has mass $m \mathrm {~kg}$ and is moving with velocity $\left[ \begin{array} { r } 5 \\ - 3 \end{array} \right] \mathrm { ms } ^ { - 1 }$.
The particle $B$ has mass 0.2 kg and is moving with velocity $\left[ \begin{array} { l } 2 \\ 3 \end{array} \right] \mathrm { ms } ^ { - 1 }$.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $m$, an expression for the total momentum of the particles.
\item The particles $A$ and $B$ collide and form a single particle $C$, which moves with velocity $\left[ \begin{array} { c } k \\ 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$, where $k$ is a constant.
\begin{enumerate}[label=(\roman*)]
\item Show that $m = 0.1$.
\item Find the value of $k$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA M1 2006 Q8 [8]}}