| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2016 |
| Session | Specimen |
| Marks | 12 |
| Topic | Impulse and momentum (advanced) |
| Type | Find coefficient of restitution |
| Difficulty | Challenging +1.8 This is an advanced mechanics problem requiring application of conservation of momentum in 2D, Newton's experimental law for oblique collisions, and energy considerations. It demands systematic vector decomposition, algebraic manipulation of multiple simultaneous equations, and geometric insight about perpendicular final velocities. While the techniques are standard for Further Maths mechanics, the multi-constraint setup and algebraic complexity place it well above average difficulty. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03i Coefficient of restitution: e6.03k Newton's experimental law: direct impact |
From Question 10 in the mark scheme:
**(i)** Let $u$ denote speed of sphere $Q$ before impact, $v_1$ and $v_2$ the speeds of spheres $Q$ and $P$, respectively, after impact and $\alpha$ the angle between $Q$'s initial direction of motion and the line of centres.
After impact, if moving perpendicularly, $Q$ moves perpendicular to line of centres and $P$ moves along line of centres. (Stated or implied) **B1**
Conservation of linear motion: $mu\cos\alpha = 0 + 3mv_2$ or $mu_x = 3mv$ **M1 A1**
Newton's experimental law: $eu\cos\alpha = v_2$ or $eu_x = v$ **A1**
$\therefore e = \frac{1}{3}$ **A1**
**(ii)** $v_1 = u\sin\alpha$ and $v_2 = \frac{1}{3}u\cos\alpha$ **B1 (both)**
Loss in kinetic energy is
$\frac{1}{2}mu^2 - \frac{1}{2}mu^2\sin^2\alpha - \frac{1}{2}\cdot 3m\frac{u^2\cos^2\alpha}{9}$ **M1 A1**
$= \frac{1}{12}mu^2$ (Or remaining kinetic energy is 5/6 of initial kinetic energy etc.) **A1**
But $\cos^2\alpha + \sin^2\alpha = 1$ (used) **M1**
$\Rightarrow \ldots \Rightarrow \sin^2\alpha = \frac{3}{4} \Rightarrow \sin\alpha = \frac{\sqrt{3}}{2} \Rightarrow \alpha = 60°$ **M1 A1**11\\
\includegraphics[max width=\textwidth, alt={}, center]{c4bbba86-2968-4247-b300-357217cf213b-5_432_949_909_557}
A smooth sphere $P$ of mass $3 m$ is at rest on a smooth horizontal table. A second smooth sphere $Q$ of mass $m$ and the same radius as $P$ is moving along the table towards $P$ and strikes it obliquely (see diagram). After the collision, the directions of motion of the two spheres are perpendicular.\\
(i) Find the coefficient of restitution.\\
(ii) Given that one-sixth of the original kinetic energy is lost as a result of the collision, find the angle between the initial direction of motion of $Q$ and the line of centres.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2016 Q11 [12]}}