| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2017 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Collision followed by wall impact |
| Difficulty | Challenging +1.2 This is a standard multi-collision momentum problem requiring systematic application of conservation of momentum and Newton's restitution law across three collisions. While it involves multiple steps and careful bookkeeping of velocities, the techniques are routine for Further Maths students and the problem structure is predictable. The 'show that' in part (i) provides scaffolding for part (ii), reducing the problem-solving demand. |
| 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) | mv + mv = mu (AEF) | |
| A B | *M1 | Use conservation of momentum (allow v + v = u) |
| Answer | Marks | Guidance |
|---|---|---|
| B A | *M1 | Use Newton’s restitution law (consistent LHS signs) |
| Answer | Marks | Guidance |
|---|---|---|
| B | A1 | Combine to find v |
| Answer | Marks | Guidance |
|---|---|---|
| B B | B1 | Verify speed w of B after collision with wall (ignore |
| Answer | Marks | Guidance |
|---|---|---|
| Total: | 4 | |
| Question | Answer | Marks |
| Answer | Marks | Guidance |
|---|---|---|
| 3(ii) | v = u / 6 | |
| A | DA1 | Find v (dependent on above *M1 *M1) |
| Answer | Marks | Guidance |
|---|---|---|
| A B B | (M1 A1 | EITHER: Equate times in terms of reqd. distance x |
| 6(d – x) = 1⋅2 d + 3⋅6 x | M1 A1) | Substitute for speeds to formulate an eqn. in x |
| Answer | Marks | Guidance |
|---|---|---|
| A B A | (M1 | OR: Find dist. x moved by A when B reaches wall |
| Answer | Marks | Guidance |
|---|---|---|
| 2 A B | M1 A1 | Find remaining time t |
| Answer | Marks | Guidance |
|---|---|---|
| A A 2 B B 2 | A1) | Find remaining distance moved by A or B |
| Answer | Marks | Guidance |
|---|---|---|
| A B A | (M1 | OR2: Find dist. x moved by A when B reaches wall |
| Answer | Marks | Guidance |
|---|---|---|
| A B A B B | M1 A1 | Equate remaining times to formulate an eqn. in x |
| 4⋅8 d – 6 x = 3⋅6 x or 1⋅8 d = 3⋅6 x | A1) | |
| x = ½ d | A1 | Find x |
| Total: | 6 |
Question 3:
--- 3(i) ---
3(i) | mv + mv = mu (AEF)
A B | *M1 | Use conservation of momentum (allow v + v = u)
A B
v – v = ⅔ u
B A | *M1 | Use Newton’s restitution law (consistent LHS signs)
v = 5u/6
B | A1 | Combine to find v
B
w = ⅓ v = 5u/18 AG
B B | B1 | Verify speed w of B after collision with wall (ignore
B
sign)
Total: | 4
Question | Answer | Marks | Guidance
--- 3(ii) ---
3(ii) | v = u / 6
A | DA1 | Find v (dependent on above *M1 *M1)
A
EITHER:
(d – x) / v = d / v + x / w (AEF)
A B B | (M1 A1 | EITHER: Equate times in terms of reqd. distance x
6(d – x) = 1⋅2 d + 3⋅6 x | M1 A1) | Substitute for speeds to formulate an eqn. in x
OR:
x = (d/v ) v = (6d/5u) u/6 = 0⋅2 d
A B A | (M1 | OR: Find dist. x moved by A when B reaches wall
A
t = (0⋅8 d) / (v + w ) = 9d/5u
2 A B | M1 A1 | Find remaining time t
2
y = v t = 0⋅3 d or y = w t = 0⋅5 d
A A 2 B B 2 | A1) | Find remaining distance moved by A or B
OR2:
x = (d/v ) v = (6d/5u) u/6 = 0⋅2 d
A B A | (M1 | OR2: Find dist. x moved by A when B reaches wall
A
(0⋅8 d – x) / v = x / w or 0⋅8 d/(v + w ) = x/w
A B A B B | M1 A1 | Equate remaining times to formulate an eqn. in x
4⋅8 d – 6 x = 3⋅6 x or 1⋅8 d = 3⋅6 x | A1)
x = ½ d | A1 | Find x
Total: | 6Two uniform small smooth spheres $A$ and $B$ have equal radii and each has mass $m$. Sphere $A$ is moving with speed $u$ on a smooth horizontal surface when it collides directly with sphere $B$ which is at rest. The coefficient of restitution between the spheres is $\frac{2}{3}$. Sphere $B$ is initially at a distance $d$ from a fixed smooth vertical wall which is perpendicular to the direction of motion of $A$. The coefficient of restitution between $B$ and the wall is $\frac{1}{3}$.
\begin{enumerate}[label=(\roman*)]
\item Show that the speed of $B$ after its collision with the wall is $\frac{5}{18}u$. [4]
\item Find the distance of $B$ from the wall when it collides with $A$ for the second time. [6]
\end{enumerate}
\hfill \mbox{\textit{CAIE FP2 2017 Q3 [10]}}