| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2012 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Oblique and successive collisions |
| Type | Energy loss in collision |
| Difficulty | Standard +0.8 This is a two-stage collision problem requiring: (1) oblique impact with a wall using coefficient of restitution and component resolution, (2) direct collision between particles with impulse-momentum calculations, and (3) finding an unknown coefficient of restitution. While the individual techniques are A-level standard, the multi-stage nature, the need to track velocity components through two collisions, and the algebraic manipulation with surds makes this moderately challenging—harder than typical mechanics questions but not requiring exceptional insight. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03f Impulse-momentum: relation6.03i Coefficient of restitution: e6.03j Perfectly elastic/inelastic: collisions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(v\cos\theta = u\cos 60° = \frac{u}{2}\) | B1 | Resolve speeds parallel to barrier |
| \(v\sin\theta = \frac{1}{3}u\sin 60° = \frac{u}{2\sqrt{3}}\) | M1 | Resolve speeds perpendicular to barrier |
| \(v^2 = u^2\left(\frac{1}{12} + \frac{1}{4}\right) = \frac{1}{3}u^2\) | A1 | Find \(v^2\) |
| \(\frac{1}{2}2m(u^2 - v^2) = \frac{2}{3} \times \frac{1}{2}2mu^2\) | M1 B1 | A.G. Relate loss of K.E. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(2mw_P = \frac{2}{3}mu(1+\sqrt{3}) - 2mv\) | M1 A1 | Find speed of \(P\) using impulse |
| \(w_P = \frac{1}{3}u\) | A.G. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(mw_Q = \frac{2}{3}mu(1+\sqrt{3}) - mu\) | M1 A1 | Find speed of \(Q\) using impulse or conservation of momentum |
| \(w_Q = \left(\frac{2}{\sqrt{3}} - \frac{1}{3}\right)u\) | (A.E.F.) | |
| \(\frac{(w_P + w_Q)}{(v+u)} = \frac{2}{(1+\sqrt{3})}\) or \(\sqrt{3}-1\) | M1 A1 | Find coefficient of restitution |
| Subtotal: 4 | Total: [11] |
## Question 4:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $v\cos\theta = u\cos 60° = \frac{u}{2}$ | B1 | Resolve speeds parallel to barrier |
| $v\sin\theta = \frac{1}{3}u\sin 60° = \frac{u}{2\sqrt{3}}$ | M1 | Resolve speeds perpendicular to barrier |
| $v^2 = u^2\left(\frac{1}{12} + \frac{1}{4}\right) = \frac{1}{3}u^2$ | A1 | Find $v^2$ |
| $\frac{1}{2}2m(u^2 - v^2) = \frac{2}{3} \times \frac{1}{2}2mu^2$ | M1 B1 | **A.G.** Relate loss of K.E. |
**Subtotal: 5**
### Question 4(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2mw_P = \frac{2}{3}mu(1+\sqrt{3}) - 2mv$ | M1 A1 | Find speed of $P$ using impulse |
| $w_P = \frac{1}{3}u$ | | **A.G.** |
**Subtotal: 2**
### Question 4(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $mw_Q = \frac{2}{3}mu(1+\sqrt{3}) - mu$ | M1 A1 | Find speed of $Q$ using impulse or conservation of momentum |
| $w_Q = \left(\frac{2}{\sqrt{3}} - \frac{1}{3}\right)u$ | | (A.E.F.) |
| $\frac{(w_P + w_Q)}{(v+u)} = \frac{2}{(1+\sqrt{3})}$ or $\sqrt{3}-1$ | M1 A1 | Find coefficient of restitution |
**Subtotal: 4 | Total: [11]**
---4 A particle $P$ of mass $2 m$, moving on a smooth horizontal plane with speed $u$, strikes a fixed smooth vertical barrier. Immediately before the collision the angle between the direction of motion of $P$ and the barrier is $60 ^ { \circ }$. The coefficient of restitution between $P$ and the barrier is $\frac { 1 } { 3 }$. Show that $P$ loses two-thirds of its kinetic energy in the collision.
Subsequently $P$ collides directly with a particle $Q$ of mass $m$ which is moving on the plane with speed $u$ towards $P$. The magnitude of the impulse acting on each particle in the collision is $\frac { 2 } { 3 } m u ( 1 + \sqrt { 3 } )$.\\
(i) Show that the speed of $P$ after this collision is $\frac { 1 } { 3 } u$.\\
(ii) Find the exact value of the coefficient of restitution between $P$ and $Q$.
\hfill \mbox{\textit{CAIE FP2 2012 Q4 [11]}}