CAIE Further Paper 3 2023 June — Question 4 8 marks

Exam BoardCAIE
ModuleFurther Paper 3 (Further Paper 3)
Year2023
SessionJune
Marks8
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Mark schemeDownload PDF ↗
TopicImpulse and momentum (advanced)
TypeOblique collision of spheres
DifficultyChallenging +1.2 This is a standard oblique collision problem requiring resolution of velocities along/perpendicular to the line of centres, application of conservation of momentum and Newton's restitution law, then finding the value of e from the constraint that spheres move in the same direction. While it involves multiple steps and careful component resolution, it follows a well-established procedure taught in Further Mechanics with no novel insight required. The 8 marks and straightforward constraint make it moderately above average difficulty.
Spec6.03k Newton's experimental law: direct impact6.03l Newton's law: oblique impacts
\includegraphics{figure_4} Two identical smooth uniform spheres \(A\) and \(B\) each have mass \(m\). The two spheres are moving on a smooth horizontal surface when they collide with speeds \(u\) and \(2u\) respectively. Immediately before the collision, \(A\)'s direction of motion makes an angle of \(30°\) with the line of centres, and \(B\)'s direction of motion is perpendicular to the line of centres (see diagram). After the collision, \(A\) and \(B\) are moving in the same direction. The coefficient of restitution between the spheres is \(e\).
  1. Find the value of \(e\). [5]
  2. Find the loss in the total kinetic energy of the spheres as a result of the collision. [3]
Question 4:
AnswerMarks
4Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored (isw).
AnswerMarks
4(a)Let speeds of A and B along line of centres after collision be V and V
A B
V V ucos30 (1)
AnswerMarks Guidance
A BM1 Allow sign errors, allow missing m.
V V eucos30 (2)
AnswerMarks Guidance
A BM1 Signs on LHS must be consistent with (1).
Speeds perpendicular to line of centres after collision are usin30 and 2u
V V
Moving in same direction, so A  B (3)
AnswerMarks Guidance
usin30 2uB1 SOI
V 4V
B A
Use V 4V in (1): 5V ucos30
B A A
From (2): 3V eucos30 then
A
AnswerMarks Guidance
Combine to find equation in e only.M1 A complete method to find equation in e only
e 3
AnswerMarks Guidance
5A1
QuestionAnswer Marks
4(a)Alternative method for question 4(a)
Let speeds of A and B along line of centres after collision be V and V
A B
V V ucos30 (1)
AnswerMarks Guidance
A BM1 Allow sign errors, allow missing m.
V V eucos30 (2)
AnswerMarks Guidance
A BM1 Signs on LHS must be consistent with (1).
Speeds perpendicular to line of centres after collision are usin30 and 2u
V V
Moving in same direction, so A  B (3)
AnswerMarks Guidance
usin30 2uB1 SOI
V 4V
B A
1 1
Solve (1) and (2): V  u1ecos30, V  u1ecos30
A 2 B 2
AnswerMarks Guidance
Substitute in (3) to find equation in e only .M1 u 4u
Note: V  3, V  3
A 10 B 10
e 3
AnswerMarks
5A1
5
AnswerMarks Guidance
4(b)KE after = 1 2 m    V A 2     u 2    2    1 2 m (2u2 V B 2)
 B1 Correct expression for KE for one of the spheres,
after collision, with both components.
7 56
KE for A after = mu2 or KE for B after = mu2
50 25
9 6
or KE loss for A = mu2 or KE gain for B = mu2
AnswerMarks Guidance
25 25B1 119
Implied by total KE after = mu2.
50
3

Total loss in KE = mu2

AnswerMarks Guidance
25B1 1
m2u2
Term may be omitted from KE of B
2
before and after.
3
AnswerMarks Guidance
QuestionAnswer Marks 
Question 4:
4 | Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored (isw).
--- 4(a) ---
4(a) | Let speeds of A and B along line of centres after collision be V and V
A B
V V ucos30 (1)
A B | M1 | Allow sign errors, allow missing m.
V V eucos30 (2)
A B | M1 | Signs on LHS must be consistent with (1).
Speeds perpendicular to line of centres after collision are usin30 and 2u
V V
Moving in same direction, so A  B (3)
usin30 2u | B1 | SOI
V 4V
B A
Use V 4V in (1): 5V ucos30
B A A
From (2): 3V eucos30 then
A
Combine to find equation in e only. | M1 | A complete method to find equation in e only
e 3
5 | A1
Question | Answer | Marks | Guidance
4(a) | Alternative method for question 4(a)
Let speeds of A and B along line of centres after collision be V and V
A B
V V ucos30 (1)
A B | M1 | Allow sign errors, allow missing m.
V V eucos30 (2)
A B | M1 | Signs on LHS must be consistent with (1).
Speeds perpendicular to line of centres after collision are usin30 and 2u
V V
Moving in same direction, so A  B (3)
usin30 2u | B1 | SOI
V 4V
B A
1 1
Solve (1) and (2): V  u1ecos30, V  u1ecos30
A 2 B 2
Substitute in (3) to find equation in e only . | M1 | u 4u
Note: V  3, V  3
A 10 B 10
e 3
5 | A1
5
--- 4(b) ---
4(b) | KE after = 1 2 m    V A 2     u 2    2    1 2 m (2u2 V B 2)
  | B1 | Correct expression for KE for one of the spheres,
after collision, with both components.
7 56
KE for A after = mu2 or KE for B after = mu2
50 25
9 6
or KE loss for A = mu2 or KE gain for B = mu2
25 25 | B1 | 119
Implied by total KE after = mu2.
50
3
Total loss in KE = mu2
25 | B1 | 1
m2u2
Term may be omitted from KE of B
2
before and after.
3
Question | Answer | Marks | Guidance
\includegraphics{figure_4}

Two identical smooth uniform spheres $A$ and $B$ each have mass $m$. The two spheres are moving on a smooth horizontal surface when they collide with speeds $u$ and $2u$ respectively. Immediately before the collision, $A$'s direction of motion makes an angle of $30°$ with the line of centres, and $B$'s direction of motion is perpendicular to the line of centres (see diagram). After the collision, $A$ and $B$ are moving in the same direction. The coefficient of restitution between the spheres is $e$.

\begin{enumerate}[label=(\alph*)]
\item Find the value of $e$. [5]
\item Find the loss in the total kinetic energy of the spheres as a result of the collision. [3]
\end{enumerate}

\hfill \mbox{\textit{CAIE Further Paper 3 2023 Q4 [8]}}
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