| Exam Board | AQA |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2008 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Oblique and successive collisions |
| Type | Sphere rebounds off fixed wall obliquely |
| Difficulty | Standard +0.8 This M3 collision problem requires understanding of oblique impacts with walls, applying coefficient of restitution in the normal direction while preserving tangential velocity, then using the perpendicularity condition to find the angle. The multi-step reasoning (resolving velocities, applying e=3/4, using the geometric constraint, then impulse-momentum) and the non-standard perpendicular trajectory condition make this moderately challenging, though the techniques are all standard M3 material. |
| Spec | 6.03e Impulse: by a force6.03f Impulse-momentum: relation6.03i Coefficient of restitution: e6.03k Newton's experimental law: direct impact |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Parallel to wall: \(u\cos\alpha = v\sin\alpha\) | M1 | |
| Perpendicular to wall (Law of Restitution): \(\frac{v\cos\alpha}{u\sin\alpha} = \frac{3}{4}\) | M1 | |
| \(\frac{v\cos\alpha}{v\tan\alpha\sin\alpha} = \frac{3}{4}\) | m1 | Dependent on both M1s |
| \(\frac{\cos^2\alpha}{\sin^2\alpha} = \frac{3}{4}\) | m1 | Dependent on both M1s |
| \(\tan^2\alpha = \frac{4}{3}\) | ||
| \(\tan\alpha = \frac{2}{\sqrt{3}}\) | A1 | 5 marks Answer given |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(v = \frac{u}{\tan\alpha}\) | M1 | |
| \(v = \frac{\sqrt{3}}{2}u\) or \(0.866u\) | A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Magnitude of Impulse = Change in momentum perpendicular to the wall | M1 | |
| \(= 0.2 \times v\cos\alpha - (-0.2 \times 4\sin\alpha)\) | A1 A1 | |
| \(= 0.2 \times \frac{\sqrt{3}}{2} \times 4\cos\alpha + 0.2 \times 4\sin\alpha\) | m1 | |
| \(= 1.06\) Ns | A1F | |
| Average Force \(= \frac{1.06}{0.1} = 10.6\) N | A1F | 6 marks |
## Question 6(a):
| Working/Answer | Marks | Guidance |
|---|---|---|
| Parallel to wall: $u\cos\alpha = v\sin\alpha$ | M1 | |
| Perpendicular to wall (Law of Restitution): $\frac{v\cos\alpha}{u\sin\alpha} = \frac{3}{4}$ | M1 | |
| $\frac{v\cos\alpha}{v\tan\alpha\sin\alpha} = \frac{3}{4}$ | m1 | Dependent on both M1s |
| $\frac{\cos^2\alpha}{\sin^2\alpha} = \frac{3}{4}$ | m1 | Dependent on both M1s |
| $\tan^2\alpha = \frac{4}{3}$ | | |
| $\tan\alpha = \frac{2}{\sqrt{3}}$ | A1 | **5 marks** Answer given |
## Question 6(b):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $v = \frac{u}{\tan\alpha}$ | M1 | |
| $v = \frac{\sqrt{3}}{2}u$ or $0.866u$ | A1 | **2 marks** |
## Question 6(c):
| Working/Answer | Marks | Guidance |
|---|---|---|
| Magnitude of Impulse = Change in momentum perpendicular to the wall | M1 | |
| $= 0.2 \times v\cos\alpha - (-0.2 \times 4\sin\alpha)$ | A1 A1 | |
| $= 0.2 \times \frac{\sqrt{3}}{2} \times 4\cos\alpha + 0.2 \times 4\sin\alpha$ | m1 | |
| $= 1.06$ Ns | A1F | |
| Average Force $= \frac{1.06}{0.1} = 10.6$ N | A1F | **6 marks** |
---6 A small smooth ball of mass $m$, moving on a smooth horizontal surface, hits a smooth vertical wall and rebounds. The coefficient of restitution between the wall and the ball is $\frac { 3 } { 4 }$.
Immediately before the collision, the ball has velocity $u$ and the angle between the ball's direction of motion and the wall is $\alpha$. The ball's direction of motion immediately after the collision is at right angles to its direction of motion before the collision, as shown in the diagram.\\
\includegraphics[max width=\textwidth, alt={}, center]{eed9842d-cd89-4eb7-b5ba-9380971be196-4_483_344_657_854}
\begin{enumerate}[label=(\alph*)]
\item Show that $\tan \alpha = \frac { 2 } { \sqrt { 3 } }$.
\item Find, in terms of $u$, the speed of the ball immediately after the collision.
\item The force exerted on the ball by the wall acts for 0.1 seconds.
Given that $m = 0.2 \mathrm {~kg}$ and $u = 4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, find the average force exerted by the wall on the ball.
\end{enumerate}
\hfill \mbox{\textit{AQA M3 2008 Q6 [13]}}