| Exam Board | AQA |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2007 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Collision followed by wall impact |
| Difficulty | Standard +0.3 This is a standard two-stage collision problem requiring conservation of momentum and Newton's restitution law, followed by a straightforward verification that spheres meet again. The techniques are routine for M3 level with clear structure, though it requires careful sign conventions and multiple steps across two collision events. |
| Spec | 6.03b 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 |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Conservation of momentum: \(0.3(3) - 0.2(2) = 0.3v_A + 0.2v_B\) | M1A1 | |
| \(3v_A + 2v_B = 5\) ... (1) | ||
| Newton's experimental law: \(0.8 = \frac{v_B - v_A}{5}\) | M1 | |
| \(v_B - v_A = 4\) ... (2) | A1 | For both (1) and (2) |
| Solving (1) and (2) | m1 | Dependent on both M1s |
| \(v_B = 3.4\), \(v_A = -0.6\) | A1F | For both solutions; Total: 6 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(0.7 = \frac{v}{3.4}\) | M1 | |
| \(v = 2.38\) | A1F | |
| Speed of \(B\) \((2.38) >\) Speed of \(A\) \((0.6)\) \(\therefore B\) collides again with \(A\) | E1 | Cannot be gained without A1F; Total: 3 marks |
## Question 4:
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Conservation of momentum: $0.3(3) - 0.2(2) = 0.3v_A + 0.2v_B$ | M1A1 | |
| $3v_A + 2v_B = 5$ ... (1) | | |
| Newton's experimental law: $0.8 = \frac{v_B - v_A}{5}$ | M1 | |
| $v_B - v_A = 4$ ... (2) | A1 | For both (1) and (2) |
| Solving (1) and (2) | m1 | Dependent on both M1s |
| $v_B = 3.4$, $v_A = -0.6$ | A1F | For both solutions; Total: 6 marks |
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $0.7 = \frac{v}{3.4}$ | M1 | |
| $v = 2.38$ | A1F | |
| Speed of $B$ $(2.38) >$ Speed of $A$ $(0.6)$ $\therefore B$ collides again with $A$ | E1 | Cannot be gained without A1F; Total: 3 marks |4 Two small smooth spheres, $A$ and $B$, of equal radii have masses 0.3 kg and 0.2 kg respectively. They are moving on a smooth horizontal surface directly towards each other with speeds $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ respectively when they collide. The coefficient of restitution between $A$ and $B$ is 0.8 .
\begin{enumerate}[label=(\alph*)]
\item Find the speeds of $A$ and $B$ immediately after the collision.
\item Subsequently, $B$ collides with a fixed smooth vertical wall which is at right angles to the path of the sphere. The coefficient of restitution between $B$ and the wall is 0.7 .
Show that $B$ will collide again with $A$.
\end{enumerate}
\hfill \mbox{\textit{AQA M3 2007 Q4 [9]}}