| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2009 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Oblique and successive collisions |
| Type | Sphere rebounds off fixed wall obliquely |
| Difficulty | Challenging +1.2 This is a multi-part mechanics problem requiring conservation of momentum, coefficient of restitution, and oblique impact analysis. While it involves several steps and the final part requires showing a specific angle result, the techniques are standard for Further Maths mechanics: applying conservation laws, resolving velocities parallel/perpendicular to the wall, and using the restitution formula. The constraint that A comes to rest simplifies the algebra considerably. More challenging than a basic C1 question but well within the scope of routine Further Maths problem-solving. |
| Spec | 6.03a Linear momentum: p = mv6.03b Conservation of momentum: 1D two particles6.03j Perfectly elastic/inelastic: collisions6.03k Newton's experimental law: direct impact |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(v = e_1 u \leq u\ [or < u]\) A.G. | M1 A1 | Find max. speed of \(B\) using elasticity. Part mark: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(m_1 u = m_2 v \leq m_2 u\) A.G. | M1 A1 | Use conservation of momentum. Part mark: 2 |
| \(V\sin\alpha = ev\sin 60°\) *or* \(ev\sqrt{3}/2\) | M1 | Equate speeds normal to wall |
| \(V\cos\alpha = v\cos 60°\) *or* \(v/2\) | M1 | Equate speeds parallel to wall |
| \(V^2 = v^2(e^2\sin^2 60° + \cos^2 60°)\) | M1 | Eliminate \(\alpha\) by squaring and adding |
| \(\tfrac{1}{2}mV^2 = \tfrac{1}{3}(\tfrac{1}{2}mv^2)\) | B1 | Relate KEs |
| \(e^2 = (\tfrac{1}{3} - \tfrac{1}{4})/\tfrac{3}{4} = 1/9\), \(e = \tfrac{1}{3}\) | M1 A1 | Hence eliminate speeds to find \(e\) |
| \(\tan\alpha = e\tan 60° = 1/\sqrt{3}\), \(\alpha = 30°\) A.G. | M1 A1 | Show that B leaves wall at 30°. Part mark: 8. Total: [12] |
## Question 5:
### Part (i):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $v = e_1 u \leq u\ [or < u]$ **A.G.** | M1 A1 | Find max. speed of $B$ using elasticity. Part mark: 2 |
### Part (ii):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $m_1 u = m_2 v \leq m_2 u$ **A.G.** | M1 A1 | Use conservation of momentum. Part mark: 2 |
| $V\sin\alpha = ev\sin 60°$ *or* $ev\sqrt{3}/2$ | M1 | Equate speeds normal to wall |
| $V\cos\alpha = v\cos 60°$ *or* $v/2$ | M1 | Equate speeds parallel to wall |
| $V^2 = v^2(e^2\sin^2 60° + \cos^2 60°)$ | M1 | Eliminate $\alpha$ by squaring and adding |
| $\tfrac{1}{2}mV^2 = \tfrac{1}{3}(\tfrac{1}{2}mv^2)$ | B1 | Relate KEs |
| $e^2 = (\tfrac{1}{3} - \tfrac{1}{4})/\tfrac{3}{4} = 1/9$, $e = \tfrac{1}{3}$ | M1 A1 | Hence eliminate speeds to find $e$ |
| $\tan\alpha = e\tan 60° = 1/\sqrt{3}$, $\alpha = 30°$ **A.G.** | M1 A1 | Show that B leaves wall at 30°. Part mark: 8. Total: **[12]** |
---5 Two spheres $A$ and $B$, of equal radius, have masses $m _ { 1 }$ and $m _ { 2 }$ respectively. They lie at rest on a smooth horizontal plane. Sphere $A$ is projected directly towards sphere $B$ with speed $u$ and, as a result of the collision, $A$ is brought to rest. Show that\\
(i) the speed of $B$ immediately after the collision cannot exceed $u$,\\
(ii) $m _ { 1 } \leqslant m _ { 2 }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{15ed1dfc-8188-4e20-9c0b-ce31af35f0b6-3_273_611_1745_767}
After the collision, $B$ hits a smooth vertical wall which is at an angle of $60 ^ { \circ }$ to the direction of motion of $B$ (see diagram). In the impact with the wall $B$ loses $\frac { 2 } { 3 }$ of its kinetic energy. Find the coefficient of restitution between $B$ and the wall and show that the direction of motion of $B$ turns through $90 ^ { \circ }$.
\hfill \mbox{\textit{CAIE FP2 2009 Q5 [12]}}