CAIE Further Paper 3 2022 June — Question 6 9 marks

Exam BoardCAIE
ModuleFurther Paper 3 (Further Paper 3)
Year2022
SessionJune
Marks9
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicImpulse and momentum (advanced)
TypeOblique collision of spheres
DifficultyChallenging +1.2 This is a standard oblique collision problem requiring conservation of momentum along the line of centres, Newton's law of restitution, and resolution of velocities. Part (a) is routine application of formulas (4 marks), while part (b) requires algebraic manipulation with the kinetic energy condition and given angle. The problem is methodical rather than requiring novel insight, typical of Further Mechanics questions but more straightforward than proof-based or optimization problems.
Spec6.03k Newton's experimental law: direct impact6.03l Newton's law: oblique impacts
Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(km\) respectively. The two spheres are on a horizontal surface. Sphere \(A\) is travelling with speed \(u\) towards sphere \(B\) which is at rest. The spheres collide. Immediately before the collision, the direction of motion of \(A\) makes an angle \(\alpha\) with the line of centres. The coefficient of restitution between the spheres is \(\frac{1}{2}\).
  1. Show that the speed of \(B\) after the collision is \(\frac{3u \cos \alpha}{2(1 + k)}\) and find also an expression for the speed of \(A\) along the line of centres after the collision, in terms of \(k\), \(u\) and \(\alpha\). [4]
After the collision, the kinetic energy of \(A\) is equal to the kinetic energy of \(B\).
  1. Given that \(\tan \alpha = \frac{2}{3}\), find the possible values of \(k\). [5]
Question 6:
AnswerMarks
6Recovery 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 2022
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 2022 Page 5 of 14
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks Guidance
6(a)Let v and w be speeds after collision:
mvkmwmucosM1 Momentum along line of centres.
1
wv ucos
AnswerMarks Guidance
2M1 NEL consistent signs.
3ucos
Add to give
AnswerMarks Guidance
21kA1 AG
Convincing working.
Substitute back or re-solve:
2kucos
v
AnswerMarks Guidance
21kA1 Accept without modulus sign.
4
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks
6(b)2kucos 2
(usin)2  
 21k 
AnswerMarks Guidance
 B1 For speed of A (SOI).
Equal KE after collision:
1 3ucos 2 1  2kucos 2
km    m(usin)2   
2  21k 2   21k  
   
 
9kcos2 41k2sin2 2k2cos2
AnswerMarks Guidance
 M1 Equate KEs.
2
Use tan :
3
 12kk2  44kk2
AnswerMarks Guidance
16 9 81kM1
25k2 85k520 leading to 5k45k130M1 Obtain quadratic and attempt to solve.
4 13
k  or
AnswerMarks
5 5A1
5
AnswerMarks Guidance
QuestionAnswer 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 2022
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 2022 Page 5 of 14
Question | Answer | Marks | Guidance
--- 6(a) ---
6(a) | Let v and w be speeds after collision:
mvkmwmucos | M1 | Momentum along line of centres.
1
wv ucos
2 | M1 | NEL consistent signs.
3ucos
Add to give
21k | A1 | AG
Convincing working.
Substitute back or re-solve:
2kucos
v
21k | A1 | Accept without modulus sign.
4
Question | Answer | Marks | Guidance
--- 6(b) ---
6(b) | 2kucos 2
(usin)2  
 21k 
  | B1 | For speed of A (SOI).
Equal KE after collision:
1 3ucos 2 1  2kucos 2
km    m(usin)2   
2  21k 2   21k  
   
 
9kcos2 41k2sin2 2k2cos2
  | M1 | Equate KEs.
2
Use tan :
3
 12kk2  44kk2
16 9 81k | M1
25k2 85k520 leading to 5k45k130 | M1 | Obtain quadratic and attempt to solve.
4 13
k  or
5 5 | A1
5
Question | Answer | Marks | Guidance
Two uniform smooth spheres $A$ and $B$ of equal radii have masses $m$ and $km$ respectively. The two spheres are on a horizontal surface. Sphere $A$ is travelling with speed $u$ towards sphere $B$ which is at rest. The spheres collide. Immediately before the collision, the direction of motion of $A$ makes an angle $\alpha$ with the line of centres. The coefficient of restitution between the spheres is $\frac{1}{2}$.

\begin{enumerate}[label=(\alph*)]
\item Show that the speed of $B$ after the collision is $\frac{3u \cos \alpha}{2(1 + k)}$ and find also an expression for the speed of $A$ along the line of centres after the collision, in terms of $k$, $u$ and $\alpha$. [4]
\end{enumerate}

After the collision, the kinetic energy of $A$ is equal to the kinetic energy of $B$.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Given that $\tan \alpha = \frac{2}{3}$, find the possible values of $k$. [5]
\end{enumerate}

\hfill \mbox{\textit{CAIE Further Paper 3 2022 Q6 [9]}}
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