| Exam Board | AQA |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2016 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Impulse and momentum (advanced) |
| Type | Oblique collision of spheres |
| Difficulty | Challenging +1.2 This is a standard M3 oblique collision problem with two stages (ball-ball, then ball-wall). Part (a) requires resolving velocities along/perpendicular to line of centres and applying conservation of momentum plus Newton's restitution law—routine for M3 students. Part (b) adds a second collision with a geometric constraint, requiring careful resolution and algebraic manipulation to find the wall's coefficient of restitution. While multi-step and requiring precision, the techniques are standard M3 fare with no novel insights needed. The 'show that' format and specific angle values simplify the algebra. Slightly above average due to length and two-collision setup, but well within expected M3 difficulty. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10d Vector operations: addition and scalar multiplication6.03b Conservation of momentum: 1D two particles6.03c Momentum in 2D: vector form6.03k Newton's experimental law: direct impact6.03l Newton's law: oblique impacts |
| Answer | Marks | Guidance |
|---|---|---|
| CLM: \(u\cos 60° = v_A + v_B\) OE | M1 A1 | M1: Four non-zero momentum terms, A1: Correct eqn. |
| Restitution: \(eu\cos 60° = v_B - v_A\) | M1 A1 | M1: Eqn using \(e\). Allow sign errors. A1: Correct eqn. |
| \(2v_B = (1 + e)u\cos 60°\) OE | A1 | A1: Correct vel of \(B\) (AG) from correct working |
| \(v_B = \frac{1}{4}u(1 + e)\) | A1 | A1: Correct vel of \(B\) (AG) from correct working |
| \(v_A = \frac{1}{4}u(1 - e)\) OE | A1 | A1: Correct vel of \(A\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(v'_A = u\cos 30°\) OE | B1 | B1: Correct perpend. comp. |
| Answer | Marks | Guidance |
|---|---|---|
| Components of \(A\) parallel & perp to the wall | M1 A1 | M1: Components of \(A\) parallel & perp to the wall. A1: Both correct |
| Components of \(B\) parallel & perp to the wall | M1 A1 | M1: Components of \(B\) parallel & perp to the wall. A1: Both correct AG above line oe needed |
| \(\frac{u\cos 30° \cos 60° - v_A \cos 30°}{v_A\cos 60° + u\cos^2 30°} = \frac{e'v_B \sin 60°}{v_B\cos 60°}\) | dM1 | dM1: Equal ratios used to create equation |
| \(u \times \frac{\sqrt{3}}{2} \times \frac{1}{2} - \frac{1}{4}u(1-e) \times \frac{\sqrt{3}}{2} \times e' \times \frac{\sqrt{3}}{2} = \frac{1}{4}u(1-e) \times \frac{1}{2} + u\left(\frac{\sqrt{3}}{2}\right)^2 = \frac{1}{2}\) | A1 | A1: Correct equation |
| \(e' = \frac{2u - u + eu}{u - eu + 6u}\) and \(e' = \frac{1 + e}{7 - e}\) | A1 | A1: CSO (AG) |
**(a)**
Along the line of centres:
| CLM: $u\cos 60° = v_A + v_B$ OE | M1 A1 | M1: Four non-zero momentum terms, A1: Correct eqn. |
| Restitution: $eu\cos 60° = v_B - v_A$ | M1 A1 | M1: Eqn using $e$. Allow sign errors. A1: Correct eqn. |
| $2v_B = (1 + e)u\cos 60°$ OE | A1 | A1: Correct vel of $B$ (AG) from correct working |
| $v_B = \frac{1}{4}u(1 + e)$ | A1 | A1: Correct vel of $B$ (AG) from correct working |
| $v_A = \frac{1}{4}u(1 - e)$ OE | A1 | A1: Correct vel of $A$ |
Perpendicular to line of centres:
| $v'_A = u\cos 30°$ OE | B1 | B1: Correct perpend. comp. |
**(b)**
| Components of $A$ parallel & perp to the wall | M1 A1 | M1: Components of $A$ parallel & perp to the wall. A1: Both correct |
| Components of $B$ parallel & perp to the wall | M1 A1 | M1: Components of $B$ parallel & perp to the wall. A1: Both correct AG above line oe needed |
| $\frac{u\cos 30° \cos 60° - v_A \cos 30°}{v_A\cos 60° + u\cos^2 30°} = \frac{e'v_B \sin 60°}{v_B\cos 60°}$ | dM1 | dM1: Equal ratios used to create equation |
| $u \times \frac{\sqrt{3}}{2} \times \frac{1}{2} - \frac{1}{4}u(1-e) \times \frac{\sqrt{3}}{2} \times e' \times \frac{\sqrt{3}}{2} = \frac{1}{4}u(1-e) \times \frac{1}{2} + u\left(\frac{\sqrt{3}}{2}\right)^2 = \frac{1}{2}$ | A1 | A1: Correct equation |
| $e' = \frac{2u - u + eu}{u - eu + 6u}$ and $e' = \frac{1 + e}{7 - e}$ | A1 | A1: CSO (AG) |
**Total: 14 marks**
---In this question use $\cos 30° = \sin 60° = \frac{\sqrt{3}}{2}$.
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 $A$ is parallel to a fixed smooth vertical wall. At the instant of collision, the line of centres of $A$ and $B$ makes an angle of $60°$ with the wall, as shown in the diagram.
\includegraphics{figure_6}
The coefficient of restitution between $A$ and $B$ is $e$.
\begin{enumerate}[label=(\alph*)]
\item Show that the speed of $B$ immediately after the collision is $\frac{1}{4}u(1 + e)$ and 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. [7 marks]
\item Subsequently, $B$ collides with the wall. After colliding with the wall, the direction of motion of $B$ is parallel to the direction of motion of $A$ after its collision with $B$.
Show that the coefficient of restitution between $B$ and the wall is $\frac{1 + e}{7 - e}$. [7 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA M3 2016 Q6 [14]}}