| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Oblique and successive collisions |
| Type | Sphere rebounds off fixed wall obliquely |
| Difficulty | Standard +0.8 This is a two-stage collision problem requiring systematic application of momentum conservation and restitution in perpendicular directions, followed by energy loss calculation. While the techniques are standard for Further Maths mechanics (resolving velocities, applying e, finding KE loss), the sequential nature with perpendicular barriers and the need to track components through both collisions elevates it above routine exercises. The angle result requires careful geometric reasoning about velocity components after two reflections. |
| Spec | 6.02d Mechanical energy: KE and PE concepts6.03b Conservation of momentum: 1D two particles6.03j Perfectly elastic/inelastic: collisions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(u\cos\alpha\) // to \(A\) and \(eu\sin\alpha \perp\) to \(A\) | M1 A1 | Find/state components after 1st collision |
| \(eu\sin\alpha\) // to \(B\) and \(eu\cos\alpha \perp\) to \(B\) | A1 | Find/state components after 2nd collision |
| \(\tan^{-1}(eu\cos\alpha / eu\sin\alpha) = \frac{1}{2}\pi - \alpha\) | A1 | Find final angle with barrier \(B\): AG |
| \(v_1\sin\beta_1 = eu\sin\alpha\) | OR: Relate angle \(\beta_1\) after 1st colln. to \(\alpha\) | |
| \(v_1\cos\beta_1 = u\cos\alpha\) | ||
| \(\tan\beta_1 = e\tan\alpha\) | M1 A1 | |
| \(v_2\sin\beta_2 = ev_1\cos\beta_1\) | Relate \(\beta_2\) after 2nd colln. to \(\beta_1\) | |
| \(v_2\cos\beta_2 = v_1\sin\beta_1\) | ||
| \(\tan\beta_2 = e/\tan\beta_1\) | A1 | |
| AG \(\tan^{-1}(e/e\tan\alpha) = \frac{1}{2}\pi - \alpha\) | A1 | Find final angle \(\beta_2\) with barrier \(B\) |
| \(\frac{1}{2}m\{u^2 - (eu\sin\alpha)^2 - (eu\cos\alpha)^2\}\) | M1 A1 | EITHER: Find total loss or gain in KE |
| \(= \frac{1}{2}m(1-e^2)u^2\) | A1 | |
| \(u_{\text{final}} = eu\) | B1 | OR: Find/state final speed |
| \(\frac{1}{2}m(u^2 - u_{\text{final}}^2) = \frac{1}{2}m(1-e^2)u^2\) | M1 A1 | Find total loss in KE |
| Part marks: 4, 3 | Total: 7 |
## Question 2:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $u\cos\alpha$ // to $A$ and $eu\sin\alpha \perp$ to $A$ | M1 A1 | Find/state components after 1st collision |
| $eu\sin\alpha$ // to $B$ and $eu\cos\alpha \perp$ to $B$ | A1 | Find/state components after 2nd collision |
| $\tan^{-1}(eu\cos\alpha / eu\sin\alpha) = \frac{1}{2}\pi - \alpha$ | A1 | Find final angle with barrier $B$: **AG** |
| $v_1\sin\beta_1 = eu\sin\alpha$ | | OR: Relate angle $\beta_1$ after 1st colln. to $\alpha$ |
| $v_1\cos\beta_1 = u\cos\alpha$ | | |
| $\tan\beta_1 = e\tan\alpha$ | M1 A1 | |
| $v_2\sin\beta_2 = ev_1\cos\beta_1$ | | Relate $\beta_2$ after 2nd colln. to $\beta_1$ |
| $v_2\cos\beta_2 = v_1\sin\beta_1$ | | |
| $\tan\beta_2 = e/\tan\beta_1$ | A1 | |
| **AG** $\tan^{-1}(e/e\tan\alpha) = \frac{1}{2}\pi - \alpha$ | A1 | Find final angle $\beta_2$ with barrier $B$ |
| $\frac{1}{2}m\{u^2 - (eu\sin\alpha)^2 - (eu\cos\alpha)^2\}$ | M1 A1 | EITHER: Find total loss or gain in KE |
| $= \frac{1}{2}m(1-e^2)u^2$ | A1 | |
| $u_{\text{final}} = eu$ | B1 | OR: Find/state final speed |
| $\frac{1}{2}m(u^2 - u_{\text{final}}^2) = \frac{1}{2}m(1-e^2)u^2$ | M1 A1 | Find total loss in KE |
**Part marks: 4, 3 | Total: 7**
---2\\
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A uniform sphere $P$ of mass $m$ is at rest on a smooth horizontal table. The sphere is projected along the table with speed $u$ and strikes a smooth vertical barrier $A$ at an acute angle $\alpha$. It then strikes another smooth vertical barrier $B$ which is at right angles to $A$ (see diagram). The coefficient of restitution between $P$ and each of the barriers is $e$. Show that the final direction of motion of $P$ makes an angle $\frac { 1 } { 2 } \pi - \alpha$ with the barrier $B$ and find the total loss in kinetic energy as a result of the two impacts. [7]
\hfill \mbox{\textit{CAIE FP2 2015 Q2 [7]}}