| Exam Board | AQA |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2007 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Oblique and successive collisions |
| Type | Oblique collision, find velocities/angles |
| Difficulty | Standard +0.3 This is a standard M3 oblique collision question requiring resolution of velocities parallel and perpendicular to the line of centres, application of Newton's experimental law (restitution), and conservation of momentum. The question guides students through each step with parts (a)-(c), and the calculations involve straightforward substitution into standard formulae with given values like cos 30° and e=2/3. While it requires multiple techniques, it follows a predictable template that M3 students practice extensively. |
| Spec | 1.05g Exact trigonometric values: for standard angles6.03b Conservation of momentum: 1D two particles6.03c Momentum in 2D: vector form6.03i Coefficient of restitution: e6.03k Newton's experimental law: direct impact |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| Conservation of momentum along line of centres: \(mu\cos30° = mv_A + mv_B\) | M1 | |
| \(v_A + v_B = \frac{\sqrt{3}}{2}u\) ——(1) | A1 | |
| Newton's experimental law: \(e = \frac{v_B - v_A}{u\cos30° - 0}\) | M1 | |
| \(v_B - v_A = \frac{\sqrt{3}}{2}ue\) ——(2) | A1 | |
| Solving (1) and (2): \(v_B = \frac{\sqrt{3}}{4}u(1+e)\) | A1 | 5 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(\perp\quad u\sin30° = \frac{1}{2}u\) | B1 | \(u\sin30\) accepted |
| \(\parallel\quad v_A = \frac{\sqrt{3}}{2}u - \frac{\sqrt{3}}{4}u(1+e)\) | M1 A1F | |
| \(v_A = \frac{\sqrt{3}}{4}u(1-e)\) | 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(\alpha = \tan^{-1}\dfrac{\frac{1}{2}u}{\frac{\sqrt{3}}{4}u\left(1-\frac{2}{3}\right)}\) | M1 A1F | |
| \(\alpha = \tan^{-1}\dfrac{6}{\sqrt{3}}\) | ||
| \(\alpha = 74°\) | A1F | 3 marks |
# Question 6:
## Part 6(a):
| Working | Marks | Guidance |
|---------|-------|---------|
| Conservation of momentum along line of centres: $mu\cos30° = mv_A + mv_B$ | M1 | |
| $v_A + v_B = \frac{\sqrt{3}}{2}u$ ——(1) | A1 | |
| Newton's experimental law: $e = \frac{v_B - v_A}{u\cos30° - 0}$ | M1 | |
| $v_B - v_A = \frac{\sqrt{3}}{2}ue$ ——(2) | A1 | |
| Solving (1) and (2): $v_B = \frac{\sqrt{3}}{4}u(1+e)$ | A1 | 5 marks | Answer given |
## Part 6(b):
| Working | Marks | Guidance |
|---------|-------|---------|
| $\perp\quad u\sin30° = \frac{1}{2}u$ | B1 | $u\sin30$ accepted |
| $\parallel\quad v_A = \frac{\sqrt{3}}{2}u - \frac{\sqrt{3}}{4}u(1+e)$ | M1 A1F | |
| $v_A = \frac{\sqrt{3}}{4}u(1-e)$ | | 3 marks | Simplification not needed |
## Part 6(c):
| Working | Marks | Guidance |
|---------|-------|---------|
| $\alpha = \tan^{-1}\dfrac{\frac{1}{2}u}{\frac{\sqrt{3}}{4}u\left(1-\frac{2}{3}\right)}$ | M1 A1F | |
| $\alpha = \tan^{-1}\dfrac{6}{\sqrt{3}}$ | | |
| $\alpha = 74°$ | A1F | 3 marks | To the nearest degree required |
---6 A smooth spherical ball, $A$, is moving with speed $u$ in a straight line on a smooth horizontal table when it hits an identical ball, $B$, which is at rest on the table.
Just before the collision, the direction of motion of $A$ makes an angle of $30 ^ { \circ }$ with the line of the centres of the two balls, as shown in the diagram.\\
\includegraphics[max width=\textwidth, alt={}, center]{daea0765-041a-4569-a535-f90fe4708313-4_362_1632_621_242}
The coefficient of restitution between $A$ and $B$ is $e$.
\begin{enumerate}[label=(\alph*)]
\item Given that $\cos 30 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }$, show that the speed of $B$ immediately after the collision is
$$\frac { \sqrt { 3 } } { 4 } u ( 1 + e )$$
\item Find, in terms of $u$ and $e$, the components of the velocity of $A$, parallel and perpendicular to the line of centres, immediately after the collision.
\item Given that $e = \frac { 2 } { 3 }$, find the angle that the velocity of $A$ makes with the line of centres immediately after the collision. Give your answer to the nearest degree.\\
(3 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA M3 2007 Q6 [11]}}