| Exam Board | AQA |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2008 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions |
| Type | Collision with vector velocities |
| Difficulty | Moderate -0.3 This is a straightforward application of conservation of momentum in two dimensions with vector notation. Students must apply the principle separately to each component and solve two simultaneous equations for U and V, then use Pythagoras for speed. While it requires careful algebraic manipulation and understanding of vector components, it's a standard M1 collision problem with no conceptual surprises or novel problem-solving required. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation6.03b Conservation of momentum: 1D two particles6.03c Momentum in 2D: vector form |
| Answer | Marks | Guidance |
|---|---|---|
| \(5\begin{bmatrix}2U/U\end{bmatrix} + 15\begin{bmatrix}V\\-1\end{bmatrix} = 20\begin{bmatrix}V\\0\end{bmatrix}\) → \(5U - 15 = 0\) → \(U = 3\) | M1, dM1, AIF (Total: 3) | Three term equation for conservation of momentum. Equation for \(U\) based on conservation of momentum. Correct value for \(U\). Deduct one mark for using weight instead of mass. |
| Answer | Marks | Guidance |
|---|---|---|
| \(30 + 15V = 20V\) → \(30 = 5V\) → \(V = \frac{30}{5} = 6\) | M1, AIF (Total: 2) | Equation for \(V\) based on conservation of momentum. Correct value for \(V\). Deduct one mark for using weight instead of mass. |
| Answer | Marks | Guidance |
|---|---|---|
| \(v = \sqrt{3^2 + 6^2} = \sqrt{45} = 3\sqrt{5} = 6.71 \text{ ms}^{-1}\) | M1, AIF (Total: 2) | Calculation of speed. Correct speed. Allow \(\sqrt{45}\). |
### Part (a)(i)
$5\begin{bmatrix}2U/U\end{bmatrix} + 15\begin{bmatrix}V\\-1\end{bmatrix} = 20\begin{bmatrix}V\\0\end{bmatrix}$ → $5U - 15 = 0$ → $U = 3$ | M1, dM1, AIF (Total: 3) | Three term equation for conservation of momentum. Equation for $U$ based on conservation of momentum. Correct value for $U$. Deduct one mark for using weight instead of mass.
### Part (a)(ii)
$30 + 15V = 20V$ → $30 = 5V$ → $V = \frac{30}{5} = 6$ | M1, AIF (Total: 2) | Equation for $V$ based on conservation of momentum. Correct value for $V$. Deduct one mark for using weight instead of mass.
### Part (b)
$v = \sqrt{3^2 + 6^2} = \sqrt{45} = 3\sqrt{5} = 6.71 \text{ ms}^{-1}$ | M1, AIF (Total: 2) | Calculation of speed. Correct speed. Allow $\sqrt{45}$.
---4 Two particles, $A$ and $B$, are moving on a horizontal plane when they collide and coalesce to form a single particle. The mass of $A$ is 5 kg and the mass of $B$ is 15 kg . Before the collision, the velocity of $A$ is $\left[ \begin{array} { c } 2 U \\ U \end{array} \right] \mathrm { ms } ^ { - 1 }$ and the velocity of $B$ is $\left[ \begin{array} { c } V \\ - 1 \end{array} \right] \mathrm { ms } ^ { - 1 }$. After the collision, the velocity of the combined particle is $\left[ \begin{array} { l } V \\ 0 \end{array} \right] \mathrm { ms } ^ { - 1 }$.
\begin{enumerate}[label=(\alph*)]
\item Find:
\begin{enumerate}[label=(\roman*)]
\item $U$;
\item $V$.
\end{enumerate}\item Find the speed of $A$ before the collision.
\end{enumerate}
\hfill \mbox{\textit{AQA M1 2008 Q4 [7]}}