| Exam Board | AQA |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2006 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Oblique and successive collisions |
| Type | Successive collisions, three particles in line |
| Difficulty | Standard +0.3 This is a standard two-collision momentum problem requiring application of conservation of momentum and Newton's restitution law twice. Part (a) is routine bookwork with equal masses, and part (b) involves algebraic manipulation to find e from given information. The setup is straightforward with no geometric complications or novel insights required. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03i Coefficient of restitution: e6.03k Newton's experimental law: direct impact |
| Answer | Marks | Guidance |
|---|---|---|
| Conservation of momentum: \(mu = mv_A + mv_B\) | M1 | |
| \(u = v_A + v_B\) | A1 | |
| Restitution: \(eu = v_B - v_A\) | M1A1 | OE |
| \(v_B = \frac{1}{2}u(1 + e)\) | A1F | OE |
| 5 |
| Answer | Marks | Guidance |
|---|---|---|
| \(mv_B = mw_B + 2m\frac{3u}{8}\) | M1A1 | OE |
| \(ev_B = \frac{3u}{8} - w_B\) | M1A1 | OE |
| Elimination of \(w_B\) | m1 | Dependent on both M1s |
| \(4e^2 + 8e - 5 = 0\) | A1F | Simplified quadratic equation in \(e\) only |
| \(e = \frac{1}{2}\) | A1F | Stated as the only value \((0 < e < 1\) for follow through\()\) |
| 7 | ||
| 12 | Total |
### Part (a)
Conservation of momentum: $mu = mv_A + mv_B$ | M1 |
$u = v_A + v_B$ | A1 |
Restitution: $eu = v_B - v_A$ | M1A1 | OE
$v_B = \frac{1}{2}u(1 + e)$ | A1F | OE
| | **5** |
### Part (b)
$mv_B = mw_B + 2m\frac{3u}{8}$ | M1A1 | OE
$ev_B = \frac{3u}{8} - w_B$ | M1A1 | OE
Elimination of $w_B$ | m1 | Dependent on both M1s
$4e^2 + 8e - 5 = 0$ | A1F | Simplified quadratic equation in $e$ only
$e = \frac{1}{2}$ | A1F | Stated as the only value $(0 < e < 1$ for follow through$)$
| | **7** |
| | **12** | **Total** |2 Three smooth spheres $A , B$ and $C$ of equal radii and masses $m , m$ and $2 m$ respectively lie at rest on a smooth horizontal table. The centres of the spheres lie in a straight line with $B$ between $A$ and $C$. The coefficient of restitution between any two spheres is $e$.
The sphere $A$ is projected directly towards $B$ with speed $u$ and collides with $B$.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $u$ and $e$, the speed of $B$ immediately after the impact between $A$ and $B$.
\item The sphere $B$ subsequently collides with $C$. The speed of $C$ immediately after this collision is $\frac { 3 } { 8 } u$. Find the value of $e$.
\end{enumerate}
\hfill \mbox{\textit{AQA M3 2006 Q2 [12]}}