| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2010 |
| Session | June |
| Marks | 11 |
| Topic | Oblique and successive collisions |
| Type | Oblique collision, find velocities/angles |
| Difficulty | Challenging +1.8 This is a challenging multi-stage oblique collision problem requiring momentum and restitution equations in two dimensions, followed by geometric reasoning about parallel motion after wall impact. It demands careful coordinate decomposition, systematic application of collision principles across two impacts, and algebraic manipulation to find the angle, but follows standard Further Maths mechanics techniques without requiring novel insight. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03k Newton's experimental law: direct impact6.03l Newton's law: oblique impacts |
**(i)** Along the line of centres:
CLM: $mu\cos\alpha = mv_A + 2mv_B$ where $u$ is $A$'s initial speed. — M1A1
NEL: $-0.5u\cos\alpha = v_A - v_b$ — M1A1
Solving simultaneously: $v_A = 0$ and $v_B = 0.5u\cos\alpha$ — M1A1 [6]
**(ii)** After $B$'s collision with the wall it has velocity components:
$0.5u\cos^2\alpha$ parallel to the wall
$0.25u\sin\alpha\cos\alpha$ perpendicular to the wall. — B1 (both)
Since $A$ and $B$ are moving parallel to each other:
$\tan\alpha = \frac{0.5u\cos^2\alpha}{0.25u\sin\alpha\cos\alpha}$ or equivalent — M1A1
$\tan^2\alpha = 2 \Rightarrow \alpha = 54.7°$ — M1A1 [5]
**[Total: 11]**6 Two smooth spheres, $A$ and $B$, have masses $m$ and $2 m$ respectively and equal radii. Sphere $B$ is at rest on a smooth horizontal floor. Sphere $A$ is projected with speed $u$ along the floor in a direction parallel to a smooth vertical wall and strikes $B$ obliquely. Subsequently $B$ strikes the wall at an angle $\alpha$ with the wall. The coefficient of restitution between $A$ and $B$ and between $B$ and the wall is 0.5. After $B$ has struck the wall, $A$ and $B$ are moving parallel to each other.\\
(i) Write down a momentum equation and a restitution equation along the line of centres for the impact between $A$ and $B$. Hence find the components of velocity of $A$ and $B$ in this direction after this first impact.\\
(ii) Find the value of $\alpha$, giving your answer in degrees.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2010 Q6 [11]}}