| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2019 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Range of coefficient of restitution |
| Difficulty | Standard +0.8 This is a multi-stage collision problem requiring conservation of momentum, Newton's law of restitution, and careful analysis of inequality conditions to prevent further collisions. Part (i) is routine application of standard formulas, but part (ii) requires setting up and solving multiple inequalities involving the coefficient of restitution across two successive collisions, which demands systematic algebraic manipulation and physical insight about relative velocities—significantly above average difficulty for A-level. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03i Coefficient of restitution: e6.03k Newton's experimental law: direct impact |
| Answer | Marks | Guidance |
|---|---|---|
| 3(i) | 5mv + 5mv = 5mu [v + v = u] and v – v = e u | |
| A B A B B A | M1 | Use consvn. of momentum for A and B and use Newton’s |
| Answer | Marks | Guidance |
|---|---|---|
| A | A1 | Combine to verify speed of A. AG |
| Answer | Marks | Guidance |
|---|---|---|
| B | A1 | Find speed of B |
| Answer | Marks |
|---|---|
| 3(ii) | 5mv ′ + 3mv = 5mv [5v ′ + 3v = 5v ] |
| Answer | Marks | Guidance |
|---|---|---|
| C B B | M1 | Use consvn. of momentum for B and C and use Newton’s |
| Answer | Marks | Guidance |
|---|---|---|
| B B C B | A1 | Combine to find v ′ (v not reqd as B, C cannot collide again) |
| Answer | Marks | Guidance |
|---|---|---|
| 2 2 | M1 | Find condition on e using v ⩽ v ′ |
| Answer | Marks | Guidance |
|---|---|---|
| 3e2 – 10e + 3 ⩽ 0 | A1 | Simplify to a quadratic inequality |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | A1 | Solve to give a lower bound on e |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | A1 | Non-strict inequality |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 3:
--- 3(i) ---
3(i) | 5mv + 5mv = 5mu [v + v = u] and v – v = e u
A B A B B A | M1 | Use consvn. of momentum for A and B and use Newton’s
restitution law with consistent LHS signs. AEF
v = ½ (1 – e) u
A | A1 | Combine to verify speed of A. AG
v = ½ (1 + e) u
B | A1 | Find speed of B
3
--- 3(ii) ---
3(ii) | 5mv ′ + 3mv = 5mv [5v ′ + 3v = 5v ]
B C B B C B
v – v ′ = ev
C B B | M1 | Use consvn. of momentum for B and C and use Newton’s
restitution law with consistent LHS signs. AEF
v ′ = (1/8) (5 – 3e) v [v = (1/8) (5 + 5e) v ]
B B C B | A1 | Combine to find v ′ (v not reqd as B, C cannot collide again)
B C
1(1 – e) u ⩽ (1/8) (5 – 3e) × 1 (1 + e) u
2 2 | M1 | Find condition on e using v ⩽ v ′
A B
3e2 – 10e + 3 ⩽ 0 | A1 | Simplify to a quadratic inequality
1 ⩽ e
3 | A1 | Solve to give a lower bound on e
1 ⩽ e ⩽ 1
3 | A1 | Non-strict inequality
6
Question | Answer | Marks | GuidanceThree uniform small spheres $A$, $B$ and $C$ have equal radii and masses $5m$, $5m$ and $3m$ respectively. The spheres are at rest on a smooth horizontal surface, in a straight line, with $B$ between $A$ and $C$. The coefficient of restitution between each pair of spheres is $e$. Sphere $A$ is projected directly towards $B$ with speed $u$.
\begin{enumerate}[label=(\roman*)]
\item Show that the speed of $A$ after its collision with $B$ is $\frac{1}{2}u(1 - e)$ and find the speed of $B$.
[3]
\end{enumerate}
Sphere $B$ now collides with sphere $C$. Subsequently there are no further collisions between any of the spheres.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Find the set of possible values of $e$.
[6]
\end{enumerate}
\hfill \mbox{\textit{CAIE FP2 2019 Q3 [9]}}