| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2016 |
| Session | Specimen |
| Marks | 12 |
| Topic | Oblique and successive collisions |
| Type | Oblique collision, direction deflected given angle |
| Difficulty | Challenging +1.8 This is a challenging oblique collision problem requiring conservation of momentum in two directions, Newton's experimental law, the perpendicularity condition, and energy considerations. It demands systematic application of multiple principles and algebraic manipulation to find both the coefficient of restitution and the angle, going beyond standard textbook exercises but using well-established techniques for Further Maths mechanics. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03i Coefficient of restitution: e6.03j Perfectly elastic/inelastic: collisions6.03k Newton's experimental law: direct impact |
**(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**10\\
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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 Q10 [12]}}