| Exam Board | AQA |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2008 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Oblique and successive collisions |
| Type | Oblique collision, vector velocity form |
| Difficulty | Standard +0.3 This is a standard M3 oblique collision question requiring conservation of momentum (straightforward vector arithmetic), angle calculation using dot product, impulse from momentum change, and recognizing that impulse direction equals line of centres. All techniques are routine for this topic with no novel insight required, making it slightly easier than average. |
| Spec | 1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication6.03b Conservation of momentum: 1D two particles6.03c Momentum in 2D: vector form6.03e Impulse: by a force6.03f Impulse-momentum: relation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| CLM: \(m(4\mathbf{i}+3\mathbf{j})+2m(-2\mathbf{i}+2\mathbf{j}) = mv+2m(\mathbf{i}+\mathbf{j})\) | M1 | |
| \(7\mathbf{j} = v+(2\mathbf{i}+2\mathbf{j})\) | ||
| \(v = -2\mathbf{i}+5\mathbf{j}\) | A2,1,0 | A1 for one slip. Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Angle with \(\mathbf{j}\) direction: \(A\): \(\tan^{-1}\frac{2}{5}=21.8°\) | OE in \(\mathbf{i}\) direction | |
| \(B\): \(\tan^{-1}\frac{1}{1}=45°\) | M1 | M1 for two inverse tan and addition of angles |
| The angle \(= 21.8°+45°=67°\) | A1F | AWRT |
| Alternative: \((-2\mathbf{i}+5\mathbf{j})\cdot(\mathbf{i}+\mathbf{j})=\sqrt{29}\times\sqrt{2}\cos\theta\) | (M1) | Not in specification |
| \(\cos\theta = \frac{3}{\sqrt{58}}\) | (A1) | |
| \(\theta = 67°\) | (A1F) | awrt. Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| The impulse \(=\) Gain in momentum of \(A\) | M1 | |
| \(= m(-2\mathbf{i}+5\mathbf{j})-m(4\mathbf{i}+3\mathbf{j})\) | A1F | |
| \(= -6m\mathbf{i}+2m\mathbf{j}\) | A1F | Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(-3\mathbf{i}+\mathbf{j}\) or any scalar multiple of \(-3\mathbf{i}+\mathbf{j}\) | B1 | Total: 1 |
## Question 4:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| CLM: $m(4\mathbf{i}+3\mathbf{j})+2m(-2\mathbf{i}+2\mathbf{j}) = mv+2m(\mathbf{i}+\mathbf{j})$ | M1 | |
| $7\mathbf{j} = v+(2\mathbf{i}+2\mathbf{j})$ | | |
| $v = -2\mathbf{i}+5\mathbf{j}$ | A2,1,0 | A1 for one slip. **Total: 3** |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Angle with $\mathbf{j}$ direction: $A$: $\tan^{-1}\frac{2}{5}=21.8°$ | | OE in $\mathbf{i}$ direction |
| $B$: $\tan^{-1}\frac{1}{1}=45°$ | M1 | M1 for two inverse tan and addition of angles |
| The angle $= 21.8°+45°=67°$ | A1F | AWRT |
| **Alternative:** $(-2\mathbf{i}+5\mathbf{j})\cdot(\mathbf{i}+\mathbf{j})=\sqrt{29}\times\sqrt{2}\cos\theta$ | (M1) | Not in specification |
| $\cos\theta = \frac{3}{\sqrt{58}}$ | (A1) | |
| $\theta = 67°$ | (A1F) | awrt. **Total: 3** |
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| The impulse $=$ Gain in momentum of $A$ | M1 | |
| $= m(-2\mathbf{i}+5\mathbf{j})-m(4\mathbf{i}+3\mathbf{j})$ | A1F | |
| $= -6m\mathbf{i}+2m\mathbf{j}$ | A1F | **Total: 3** |
### Part (d):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $-3\mathbf{i}+\mathbf{j}$ or any scalar multiple of $-3\mathbf{i}+\mathbf{j}$ | B1 | **Total: 1** |
---4 Two smooth spheres, $A$ and $B$, have equal radii and masses $m$ and $2 m$ respectively. The spheres are moving on a smooth horizontal plane. The sphere $A$ has velocity ( $4 \mathbf { i } + 3 \mathbf { j }$ ) when it collides with the sphere $B$ which has velocity $( - 2 \mathbf { i } + 2 \mathbf { j } )$. After the collision, the velocity of $B$ is $( \mathbf { i } + \mathbf { j } )$.
\begin{enumerate}[label=(\alph*)]
\item Find the velocity of $A$ immediately after the collision.
\item Find the angle between the velocities of $A$ and $B$ immediately after the collision.
\item Find the impulse exerted by $B$ on $A$.
\item State, as a vector, the direction of the line of centres of $A$ and $B$ when they collide.\\
(1 mark)
\end{enumerate}
\hfill \mbox{\textit{AQA M3 2008 Q4 [10]}}