| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Session | Specimen |
| Marks | 3 |
| Topic | Momentum and Collisions 1 |
| Type | Two-sphere oblique collision |
| Difficulty | Challenging +1.2 This is a standard oblique collision problem requiring resolution of velocities along and perpendicular to the line of centres, application of conservation of momentum and Newton's restitution law. While it involves multiple steps and algebraic manipulation, the method is well-established and follows a predictable pattern taught in mechanics courses. The 'show that' format provides target answers to work towards, reducing problem-solving demand. |
| Spec | 6.03k Newton's experimental law: direct impact |
**Question 2**
**(i)** Before impact speed of first ball is $u$. After impact 1st ball has components $v_1$ along line of centres and $\frac{1}{4}u$ perpendicular to line of centres, and 2nd ball has speed $v_2$ along line of centres.
Along line of centres:
CLM gives $\frac{1}{2}m\sqrt{3}u = mv_1 + mv_2$ M1A1
NEL gives $-\frac{1}{2}e\sqrt{3}u = v_1 - v_2$ M1A1
Adding $\Rightarrow \frac{1}{2}\sqrt{3}u(1-e) = 2v_1 \Rightarrow v_1 = \frac{1}{4}\sqrt{3}u(1-e)$ (AG) A1 **[5 marks]**
**(ii)** Let $\phi$ be the angle between the 1st ball's final velocity and the line of centres:
$$\tan\phi = \frac{\frac{1}{4}u}{\frac{1}{4}\sqrt{3}u(1-e)} = \frac{2\sqrt{3}}{3(1-e)}$$ B1
$$\tan\theta = \tan(\phi - 30°) = \frac{\dfrac{2\sqrt{3}}{3(1-e)} - \dfrac{\sqrt{3}}{3}}{1 + \dfrac{1}{\sqrt{3}} \times \dfrac{2\sqrt{3}}{3(1-e)}}$$ M1A1
$$= \frac{2\sqrt{3} - \sqrt{3}(1-e)}{3(1-e)+2} = \frac{\sqrt{3}(1+e)}{5-3e}$$ (AG) A1 **[4 marks]**2 A smooth uniform ball travelling along a smooth horizontal table collides with a second smooth uniform ball of the same mass and radius which is at rest on the table. At the moment of impact the line of centres makes an angle of $30 ^ { \circ }$ with the direction in which the first ball is moving. If the coefficient of restitution between the balls is $e$, show that\\
(i) the component of the first ball's velocity, along the line of centres, after the impact is
$$\frac { \sqrt { 3 } u } { 4 } ( 1 - e )$$
(ii) the first ball is deflected by the impact through an angle $\theta$, where
$$\tan \theta = \frac { ( 1 + e ) \sqrt { 3 } } { 5 - 3 e }$$
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 Q2 [3]}}