| Exam Board | AQA |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2010 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Oblique and successive collisions |
| Type | Sphere rebounds off fixed wall obliquely |
| Difficulty | Standard +0.3 This is a standard M3 collision mechanics question requiring resolution of velocities parallel and perpendicular to the wall, application of the coefficient of restitution formula, and conservation of momentum parallel to the wall. The steps are routine and well-practiced for M3 students, though it requires careful component resolution and algebraic manipulation across two parts. |
| Spec | 6.03i Coefficient of restitution: e6.03k Newton's experimental law: direct impact |
| Answer | Marks | Guidance |
|---|---|---|
| Component parallel to wall unchanged: \(4\sin\alpha = v\sin 40°\) | B1 | |
| Component perpendicular to wall, applying NEL: \(v\cos 40° = \frac{2}{3} \times 4\cos\alpha\) | M1 | |
| Dividing: \(\frac{v\sin 40°}{v\cos 40°} = \frac{4\sin\alpha}{\frac{2}{3}\times 4\cos\alpha}\) | M1 | |
| \(\tan 40° = \frac{2}{3}\tan\alpha \Rightarrow \tan\alpha = \frac{3}{2}\tan 40°\) | A1 | Shown |
| Answer | Marks |
|---|---|
| \(v\cos 40° = \frac{2}{3}\times 4\cos\alpha\) | M1 |
| \(\tan\alpha = \frac{3}{2}\tan 40° \Rightarrow \alpha = \arctan(\frac{3}{2}\tan 40°)\) | M1 |
| \(v = \frac{\frac{8}{3}\cos\alpha}{\cos 40°} \approx 3.23 \text{ m s}^{-1}\) | A1 |
# Question 5:
## Part (a):
| Component parallel to wall unchanged: $4\sin\alpha = v\sin 40°$ | B1 | |
| Component perpendicular to wall, applying NEL: $v\cos 40° = \frac{2}{3} \times 4\cos\alpha$ | M1 | |
| Dividing: $\frac{v\sin 40°}{v\cos 40°} = \frac{4\sin\alpha}{\frac{2}{3}\times 4\cos\alpha}$ | M1 | |
| $\tan 40° = \frac{2}{3}\tan\alpha \Rightarrow \tan\alpha = \frac{3}{2}\tan 40°$ | A1 | Shown |
## Part (b):
| $v\cos 40° = \frac{2}{3}\times 4\cos\alpha$ | M1 | |
| $\tan\alpha = \frac{3}{2}\tan 40° \Rightarrow \alpha = \arctan(\frac{3}{2}\tan 40°)$ | M1 | |
| $v = \frac{\frac{8}{3}\cos\alpha}{\cos 40°} \approx 3.23 \text{ m s}^{-1}$ | A1 | |
---5 A smooth sphere is moving on a smooth horizontal surface when it strikes a smooth vertical wall and rebounds.
Immediately before the impact, the sphere is moving with speed $4 \mathrm {~ms} ^ { - 1 }$ and the angle between the sphere's direction of motion and the wall is $\alpha$.
Immediately after the impact, the sphere is moving with speed $v \mathrm {~ms} ^ { - 1 }$ and the angle between the sphere's direction of motion and the wall is $40 ^ { \circ }$.
The coefficient of restitution between the sphere and the wall is $\frac { 2 } { 3 }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-14_529_250_831_909}
\begin{enumerate}[label=(\alph*)]
\item Show that $\tan \alpha = \frac { 3 } { 2 } \tan 40 ^ { \circ }$.
\item Find the value of $v$.
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-15_2484_1709_223_153}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{AQA M3 2010 Q5 [6]}}