| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2016 |
| Session | Specimen |
| Marks | 10 |
| Topic | Momentum and Collisions 1 |
| Type | Three-particle sequential collisions |
| Difficulty | Standard +0.8 This is a multi-stage collision problem requiring systematic application of conservation of momentum and Newton's restitution law across three sequential collisions, tracking positions and velocities throughout. While the individual collision calculations are standard A-level mechanics, the extended chain of reasoning, careful bookkeeping of three particles, and the need to determine relative positions at a specific later time elevate this above typical textbook exercises. |
| Spec | 6.03a Linear momentum: p = mv6.03b Conservation of momentum: 1D two particles6.03i Coefficient of restitution: e6.03j Perfectly elastic/inelastic: collisions6.03k Newton's experimental law: direct impact |
**Question 9**
(i) COM: $1 \times 14 + 2 \times 0 = U + 2V$ [B1]
NEL: $V - U = 0.5(14 - 0)$ [B1]
$U = 0\ \text{ms}^{-1}$ [B1]
$V = 7\ \text{ms}^{-1}$ [B1]
(ii) COM: $2 \times 7 + 5 \times 0 = 2U + 5V$ [B1]
NEL: $V - U = 0.5(7 - 0)$ [B1]
$U = -0.5\ \text{ms}^{-1}$ [B1]
(iii) $V = 3\ \text{ms}^{-1}$ [B1]
$B$ reaches $A$ in 2 seconds [B1]
Distance between $A$ and $C$ is $1 + 2 \times 3 = 7$ metres [B1]
**Total: 10 marks**9\\
\includegraphics[max width=\textwidth, alt={}, center]{01bd6354-3514-4dad-901b-7ecbe155b2c7-5_118_851_1265_607}
Three particles $A , B$ and $C$, having masses of $1 \mathrm {~kg} , 2 \mathrm {~kg}$ and 5 kg respectively, are placed 1 metre apart in a straight line on a smooth horizontal plane (see diagram). The particles $B$ and $C$ are initially at rest and $A$ is moving towards $B$ with speed $14 \mathrm {~ms} ^ { - 1 }$. The coefficient of restitution between each pair of particles is 0.5 .\\
(i) Find the velocity of $B$ immediately after the first impact and show that $A$ comes to rest.\\
(ii) Show that $B$ reversed direction after the impact with $C$.\\
(iii) Find the distances between $B$ and $C$ at the instant that $B$ collides with $A$ for the second time.
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2016 Q9 [10]}}