| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2012 |
| Session | June |
| Marks | 10 |
| Topic | Momentum and Collisions 1 |
| Type | Three-particle sequential collisions |
| Difficulty | Challenging +1.2 This is a sequential collision problem requiring systematic application of conservation of momentum and Newton's restitution law across multiple impacts. While it involves three collisions and tracking particle positions, each collision follows standard A-level mechanics procedures. The multi-part structure and need to track positions between collisions elevates it above routine questions, but the techniques are all standard for Further Maths mechanics. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03j Perfectly elastic/inelastic: collisions6.03k Newton's experimental law: direct impact |
**(i)** COM: $1 \times 14 + 2 \times 0 = U + 2V$ — B1
NEL: $V - U = 0.5(14 - 0)$ — B1
Solution: $U = 0$ ms$^{-1}$, — B1
$V = 7$ ms$^{-1}$ — B1 [4]
**(ii)** For impact of $B$ on $C$:
COM: $2 \times 7 + 5 \times 0 = 2U + 5V$ — B1
NEL: $V - U = 0.5(7 - 0)$ — B1
Solution: $U = -0.5$ ms$^{-1}$ — B1 [3]
**(iii)** $V = 3$ ms$^{-1}$ — B1
$B$ reaches $A$ in 2 seconds. — B1
Distance between $A$ and $C$ is $1 + 2 \times 3 = 7$ metres — B1 [3]
**Total: [10]**
*Depends on 2nd B1. SC If NEL is $V + U = \ldots$ then max B1B0B0B1. May be seen/awarded in (ii). ft their $U$. ft their $V$.*10\\
\includegraphics[max width=\textwidth, alt={}, center]{f0c32e07-f3a0-4d58-bd00-c266177ceaac-4_81_949_1283_598}
Three particles $A , B$ and $C$, having masses $1 \mathrm {~kg} , 2 \mathrm {~kg}$ 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 $14 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. 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 $B$ reverses direction after an impact with $C$.\\
(iii) Find the distance 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 2012 Q10 [10]}}