| Exam Board | CAIE |
|---|---|
| Module | Further Paper 3 (Further Paper 3) |
| Year | 2021 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Impulse and momentum (advanced) |
| Type | Oblique collision of spheres |
| Difficulty | Challenging +1.8 This is a challenging Further Maths mechanics problem requiring systematic application of conservation of momentum (in two directions), Newton's law of restitution, and energy considerations. The constraint 2cos β = cos α must be used strategically, and part (b) requires algebraic manipulation of multiple equations. However, the solution path is relatively standard for oblique collision problems once the framework is established. |
| Spec | 6.02d Mechanical energy: KE and PE concepts6.03k 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) | Along line of centres, speeds v and v |
| Answer | Marks | Guidance |
|---|---|---|
| 1 2 | M1 | Momentum (condone missing masses). |
| Answer | Marks | Guidance |
|---|---|---|
| 2 1 | M1 | Restitution. |
| Both correct, masses seen. | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| perpendicular to line of centres | A1 | AG. |
| Answer | Marks |
|---|---|
| 6(b) | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | M1 | Both components. |
| Answer | Marks |
|---|---|
| 4 | M1 |
| Answer | Marks |
|---|---|
| Simplify to equation in | M1 |
| Answer | Marks |
|---|---|
| 2 | 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
9231/31 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 13
Question | Answer | Marks | Guidance
--- 6(a) ---
6(a) | Along line of centres, speeds v and v
1 2
mv +mv = mucosα−mucosβ
1 2 | M1 | Momentum (condone missing masses).
( )
v −v =eu cosβ+ cosα
2 1 | M1 | Restitution.
Both correct, masses seen. | A1
v = 0 so A has no speed along line of centres: moves
1
perpendicular to line of centres | A1 | AG.
4
--- 6(b) ---
6(b) | 1
(v = u cosα=ucosβ )
2 2
( )
1 +( usinβ)2
KE of B after collision is m v 2
2 2
1 ( usinα)2
KE of A after collision = m
2 | M1 | Both components.
3
Add both KEs and equate to mu2
4 | M1
sinα
Simplify to equation in | M1
1 α=45°
sinα= ,
2 | A1
4
Question | Answer | Marks | Guidance\includegraphics{figure_6}
Two uniform smooth spheres A and B of equal radii each have mass $m$. The two spheres are each moving with speed $u$ on a horizontal surface when they collide. Immediately before the collision, A's direction of motion makes an angle $\alpha$ with the line of centres, and B's direction of motion makes an angle $\beta$ with the line of centres (see diagram). The coefficient of restitution between the spheres is $\frac{1}{3}$ and $2\cos\beta = \cos\alpha$.
\begin{enumerate}[label=(\alph*)]
\item Show that the direction of motion of A after the collision is perpendicular to the line of centres. [4]
\end{enumerate}
The total kinetic energy of the spheres after the collision is $\frac{3}{4}mu^2$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the value of $\alpha$. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 3 2021 Q6 [8]}}