| Exam Board | CAIE |
|---|---|
| Module | Further Paper 3 (Further Paper 3) |
| Year | 2021 |
| Session | June |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Impulse and momentum (advanced) |
| Type | Oblique collision of spheres |
| Difficulty | Standard +0.8 This is a standard oblique collision problem requiring application of conservation of momentum along the line of centres and the restitution equation. While it involves multiple parameters (m, k, θ) and requires careful component resolution, the method is well-established for Further Maths students. The 'show that' format with 3 marks indicates a straightforward derivation following standard techniques, placing it moderately above average difficulty. |
| Spec | 6.03k Newton's experimental law: direct impact |
| Answer | Marks |
|---|---|
| 6 | Recovery within working is allowed, e.g. a notation error in the working where the following line of working makes the candidate’s intent clear. |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks |
|---|---|
| 6(a) | Let velocities of A and B along line of centres after collision be v |
| Answer | Marks | Guidance |
|---|---|---|
| 1 2 | M1 | Momentum, must include m, allow cos/sin mix. |
| Answer | Marks | Guidance |
|---|---|---|
| 2 1 3 | M1 | Restitution, consistent signs, correct way up. |
| Answer | Marks | Guidance |
|---|---|---|
| Solve: v 2 = 3 ( 1+k ) | A1 | AG shown convincingly. |
| Answer | Marks |
|---|---|
| 6(b) | ( 3−k ) ucosθ |
| Answer | Marks | Guidance |
|---|---|---|
| 1 3 ( 1+k ) | B1 | Or equivalent, may be unsimplified. |
| Use velocity of A with both components. | B1 | v2+( usinθ)2 |
| Answer | Marks | Guidance |
|---|---|---|
| 2 2 2 1 10 2 | M1 | KE after = 30% KE before (all terms present). |
| Answer | Marks | Guidance |
|---|---|---|
| Substitute from part (a) and for . | M1 | Eliminate trigonometric terms, must be KE equation, in terms of |
| Answer | Marks | Guidance |
|---|---|---|
| , k2 −6k −7=0 | M1 | Obtain simplified quadratic equation in k . |
| k =7 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 6:
6 | Recovery within working is allowed, e.g. a notation error in the working where the following line of working makes the candidate’s intent clear.
PMT
9231/33 Cambridge International AS & A Level – Mark Scheme May/June 2021
PUBLISHED
Abbreviations
AEF/OE Any Equivalent Form (of answer is equally acceptable) / Or Equivalent
AG Answer Given on the question paper (so extra checking is needed to ensure that the detailed working leading to the result is valid)
CAO Correct Answer Only (emphasising that no ‘follow through’ from a previous error is allowed)
CWO Correct Working Only
ISW Ignore Subsequent Working
SOI Seen Or Implied
SC Special Case (detailing the mark to be given for a specific wrong solution, or a case where some standard marking practice is to be varied in the
light of a particular circumstance)
WWW Without Wrong Working
AWRT Answer Which Rounds To
© UCLES 2021 Page 5 of 16
Question | Answer | Marks | Guidance
--- 6(a) ---
6(a) | Let velocities of A and B along line of centres after collision be v
1
and v .
2
mv + kmv = mucosθ .
1 2 | M1 | Momentum, must include m, allow cos/sin mix.
1
v −v = ucosθ
2 1 3 | M1 | Restitution, consistent signs, correct way up.
4ucosθ
Solve: v 2 = 3 ( 1+k ) | A1 | AG shown convincingly.
3
--- 6(b) ---
6(b) | ( 3−k ) ucosθ
v =
1 3 ( 1+k ) | B1 | Or equivalent, may be unsimplified.
Use velocity of A with both components. | B1 | v2+( usinθ)2
seen.
1
( )
1 kmv 2 + 1 m v2 +( usinθ)2 = 3 × 1 mu2
2 2 2 1 10 2 | M1 | KE after = 30% KE before (all terms present).
M0 if incorrect masses.
θ
Substitute from part (a) and for . | M1 | Eliminate trigonometric terms, must be KE equation, in terms of
k only.
( 3−k )2 +16k=2 ( 1+k )2
, k2 −6k −7=0 | M1 | Obtain simplified quadratic equation in k .
k =7 | A1
6
Question | Answer | Marks | Guidance\includegraphics{figure_6}
Two uniform smooth spheres $A$ and $B$ of equal radii have masses $m$ and $km$ respectively. Sphere $A$ is moving with speed $u$ on a smooth horizontal surface when it collides with sphere $B$ which is at rest. Immediately before the collision, $A$'s direction of motion makes an angle $\theta$ with the line of centres (see diagram). The coefficient of restitution between the spheres is $\frac{1}{3}$.
\begin{enumerate}[label=(\alph*)]
\item Show that the speed of $B$ after the collision is $\frac{4u \cos \theta}{3(1 + k)}$. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 3 2021 Q6 [3]}}