| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2015 |
| Session | June |
| Marks | 7 |
| Topic | Momentum and Collisions 1 |
| Type | Particle brought to rest by collision |
| Difficulty | Moderate -0.3 This is a straightforward momentum and restitution problem requiring standard application of conservation of momentum and Newton's experimental law. Part (i) is direct substitution, part (ii) combines the two equations algebraically, and part (iii) uses the constraint 0 ≤ e ≤ 1. While it requires careful algebra across three parts, it follows a completely standard template for collision problems with no novel insight required, making it slightly easier than average. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03j Perfectly elastic/inelastic: collisions6.03k Newton's experimental law: direct impact |
**Question 9**
**(i)** C of M: $0.5u\ (+0) = (0+)\ kv$
$\therefore v = \frac{u}{2k}$
— M1. A1: c.a.o. **[2]**
**(ii)** NEL: $v(-0) = e(u-(-0))$
$\therefore \frac{u}{2k} = eu$
— M1. M1: Substitute or use *their* expression for $v$.
$\therefore e = \frac{1}{2k}$
— A1: c.a.o. **[3]**
**(iii)** $(0 \leqslant)\ e \leqslant 1$
$\therefore \frac{1}{2k} \leqslant 1 \qquad \therefore k \geqslant \frac{1}{2}$
— M1: Use of condition on $e$. A1: A.G. Convincingly shown. **[2]**9 A particle of mass 0.5 kg moving on a smooth horizontal plane with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ collides directly with another particle of mass $k \mathrm {~kg}$ (where $k$ is a constant) which is at rest. After the collision the first particle comes to rest but the second particle moves off with speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$.\\
(i) Find $v$ in terms of $k$ and $u$.\\
(ii) The coefficient of restitution between the two particles is $e$. Find $e$ in terms of $k$ only.\\
(iii) Show that $k \geqslant \frac { 1 } { 2 }$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2015 Q9 [7]}}