| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2020 |
| Session | Specimen |
| Marks | 5 |
| Topic | Impulse and momentum (advanced) |
| Type | Find coefficient of restitution |
| Difficulty | Challenging +1.8 This is an advanced oblique collision problem requiring simultaneous application of momentum conservation (in two directions), Newton's experimental law, and energy considerations. It demands systematic vector decomposition, algebraic manipulation across multiple equations, and geometric insight about perpendicular final velocities. While the individual principles are A-level standard, the synthesis and multi-constraint optimization places it well above typical mechanics questions. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03j Perfectly elastic/inelastic: collisions6.03k Newton's experimental law: direct impact |
**Question 11(a):**
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 momentum: $mu\cos\alpha = 0 + 3mv_2$ or $mu_x = 3mv$ [M1A1]
Newton's experimental law: $eu\cos\alpha = v_2$ or $eu_x = v$ [A1]
$\therefore e = \dfrac{1}{3}$ [A1]
**Total: 5 marks**
**Question 11(b):**
$v_1 = u\sin\alpha$ and $v_2 = \dfrac{1}{3}u\cos\alpha$ (both needed) [B1]
Loss in kinetic energy is $\dfrac{1}{2}mu^2 - \dfrac{1}{2}mu^2\sin^2\alpha - \dfrac{1}{2} \cdot 3m\dfrac{u^2\cos^2\alpha}{9}$ [M1A1]
$= \dfrac{1}{12}mu^2$ (Or remaining kinetic energy is 5/6 of initial kinetic energy etc.) [A1]
But $\cos^2\alpha + \sin^2\alpha = 1$ (used) [M1]
$\Rightarrow \ldots \Rightarrow \sin^2\alpha = \dfrac{3}{4} \Rightarrow \sin\alpha = \dfrac{\sqrt{3}}{2} \Rightarrow \alpha = 60°$ [M1A1]
**Total: 7 marks**11\\
\includegraphics[max width=\textwidth, alt={}, center]{f4acd242-eb78-4124-bfa2-fdecaa188690-6_438_953_264_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.
\begin{enumerate}[label=(\alph*)]
\item Find the coefficient of restitution.
\item 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.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2020 Q11 [5]}}