Question 6 - A-Level Maths

CAIE M1 2022 November — Question 6 9 marks

Exam Board	CAIE
Module	M1 (Mechanics 1)
Year	2022
Session	November
Marks	9
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Momentum and Collisions
Type	Multiple sequential collisions
Difficulty	Moderate -0.3 This is a straightforward multi-stage momentum conservation problem with standard calculations. Part (a) is routine momentum conservation (showing a given answer), part (b) applies the same principle to find an unknown mass, and part (c) requires basic kinematics to find when particles meet. All steps follow standard textbook methods with no novel insight required, making it slightly easier than average for A-level.
Spec	3.02d Constant acceleration: SUVAT formulae 6.03b Conservation of momentum: 1D two particles

Three particles \(A\), \(B\) and \(C\) of masses 0.3 kg, 0.4 kg and \(m\) kg respectively lie at rest in a straight line on a smooth horizontal plane. The distance between \(B\) and \(C\) is 2.1 m. \(A\) is projected directly towards \(B\) with speed \(2 \text{ m s}^{-1}\). After \(A\) collides with \(B\) the speed of \(A\) is reduced to \(0.6 \text{ m s}^{-1}\), still moving in the same direction.

Show that the speed of \(B\) after the collision is \(1.05 \text{ m s}^{-1}\). [2]

After the collision between \(A\) and \(B\), \(B\) moves directly towards \(C\). Particle \(B\) now collides with \(C\). After this collision, the two particles coalesce and have a combined speed of \(0.5 \text{ m s}^{-1}\).

Find \(m\). [2]

Find the time that it takes, from the instant when \(B\) and \(C\) collide, until \(A\) collides with the combined particle. [5]

Show mark scheme Show mark scheme source

Question 6:

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.

PMT

9709/42 Cambridge International AS & A Level – Mark Scheme October/November 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 22

Answer	Marks	Guidance
Question	Answer	Marks

Answer	Marks	Guidance
6(a)	0.32 +0 =0.30.6+0.4v	M1

terms. Allow sign errors.

Answer	Marks	Guidance
Speed of B = 1.05 ms−1	A1	AG Allow M1 A0 if g included with the masses.

2

Answer	Marks	Guidance
6(b)	0.41.05 +0 =(0.4+m)0.5	M1

terms. Allow sign errors.

11 m = 0.44 or

Answer	Marks	Guidance
25	A1	Allow M1 A0 if g included with the masses.

2

Answer	Marks	Guidance
Question	Answer	Marks

Answer	Marks	Guidance
6(c)	1.2 [m] or 0.9 [m]	B1

0.6 .

0.5

Answer	Marks	Guidance
0.5t	B1	1

Seen but not + at2 unless later state that a=0.

2 B0 if only 0.5t=2.1 (may see solving to find t =3.5).

Answer	Marks	Guidance
0.6t	B1	1

Seen but not + at2 unless later state that a=0.

2 Allow B2 in place of second and third B1 marks for

‘difference in speeds is 0.1 [ ms−1]’.

Distances equal so 0.6t−0.9=0.5t and solve for t

0.9 Or t =

Answer	Marks	Guidance
0.6−0.5	M1	OE Must get to ‘t=’. Allow ± their 0.9 but not ±1.2

or ± 2.1 or 1.5. Do not allow 0.6t+0.5t=0.9 .

Do not allow M1 if either or both terms include

1 + at2 unless they state a = 0.

2

Answer	Marks	Guidance
Time = 9 s	A1	CWO
Question	Answer	Marks
6(c)	Alternative method for question 6(c) using time from start of motion
1 [m] or 1.1[m]	B1
0.6T	B1	1

Seen but not + aT2 unless later state that a=0.

2

Answer	Marks	Guidance
0.5T	B1	1

Seen but not + aT2 unless later state that a=0.

2 Allow B2 in place of second and third B1 marks for

‘difference in speeds is 0.1 [ ms−1]’.

Distances equal so 0.6T−1.1=0.5T and solve for T

1.1 Or T =

Answer	Marks	Guidance
0.6−0.5	M1	OE Must get to ‘T=’. Allow ± their 1.1 but not ±1.0

or ± 2.1. Do not allow 0.6T+0.5T =1.1 .

Do not allow M1 if either or both terms include

1 + aT2 unless they state a = 0

2

Answer	Marks	Guidance
⇒ Time from BC collision = 11−2=9 s	A1	CWO
Question	Answer	Marks
6(c)	Alternative method for question 6(c) using distance travelled from time when B and C collide
1.2 [m] or 0.9 [m]	B1

d +0.9

Time taken for A is

0.6

Answer	Marks	Guidance
Or d+0.9=0.6t	B1 FT	Allow ± their 0.9 but not ±1.2 or ± 2.1 or 1.5.

d

Time taken for BC is

0.5

Answer	Marks
Or d =0.5t	B1

d+0.9 d 4.5

= d =4.5Time=

Answer	Marks	Guidance
0.6 0.5 0.5	M1	Must get to ‘t=’.

Allow ± their 0.9 but not ±1.2 or ± 2.1.

Answer	Marks	Guidance
Time = 9 s	A1	CWO

5

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
9709/42 Cambridge International AS & A Level – Mark Scheme October/November 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 22
Question | Answer | Marks | Guidance
--- 6(a) ---
6(a) | 0.32 +0 =0.30.6+0.4v | M1 | For use of conservation of momentum. Must be 3
terms. Allow sign errors.
Speed of B = 1.05 ms−1 | A1 | AG Allow M1 A0 if g included with the masses.
2
--- 6(b) ---
6(b) | 0.41.05 +0 =(0.4+m)0.5 | M1 | For use of conservation of momentum. Must be 3
terms. Allow sign errors.
11
m = 0.44 or
25 | A1 | Allow M1 A0 if g included with the masses.
2
Question | Answer | Marks | Guidance
--- 6(c) ---
6(c) | 1.2 [m] or 0.9 [m] | B1 | Must be a distance as some candidates get 1.2 from
0.6
.
0.5
0.5t | B1 | 1
Seen but not + at2 unless later state that a=0.
2
B0 if only 0.5t=2.1 (may see solving to find t =3.5).
0.6t | B1 | 1
Seen but not + at2 unless later state that a=0.
2
Allow B2 in place of second and third B1 marks for
‘difference in speeds is 0.1 [ ms−1]’.
Distances equal so 0.6t−0.9=0.5t and solve for t
0.9
Or t =
0.6−0.5 | M1 | OE Must get to ‘t=’. Allow ± their 0.9 but not ±1.2
or ± 2.1 or 1.5. Do not allow 0.6t+0.5t=0.9 .
Do not allow M1 if either or both terms include
1
+ at2 unless they state a = 0.
2
Time = 9 s | A1 | CWO
Question | Answer | Marks | Guidance
6(c) | Alternative method for question 6(c) using time from start of motion
1 [m] or 1.1[m] | B1
0.6T | B1 | 1
Seen but not + aT2 unless later state that a=0.
2
0.5T | B1 | 1
Seen but not + aT2 unless later state that a=0.
2
Allow B2 in place of second and third B1 marks for
‘difference in speeds is 0.1 [ ms−1]’.
Distances equal so 0.6T−1.1=0.5T and solve for T
1.1
Or T =
0.6−0.5 | M1 | OE Must get to ‘T=’. Allow ± their 1.1 but not ±1.0
or ± 2.1. Do not allow 0.6T+0.5T =1.1 .
Do not allow M1 if either or both terms include
1
+ aT2 unless they state a = 0
2
⇒ Time from BC collision = 11−2=9 s | A1 | CWO
Question | Answer | Marks | Guidance
6(c) | Alternative method for question 6(c) using distance travelled from time when B and C collide
1.2 [m] or 0.9 [m] | B1
d +0.9
Time taken for A is
0.6
Or d+0.9=0.6t | B1 FT | Allow ± their 0.9 but not ±1.2 or ± 2.1 or 1.5.
d
Time taken for BC is
0.5
Or d =0.5t | B1
d+0.9 d 4.5
= d =4.5Time=
0.6 0.5 0.5 | M1 | Must get to ‘t=’.
Allow ± their 0.9 but not ±1.2 or ± 2.1.
Time = 9 s | A1 | CWO
5
Question | Answer | Marks | Guidance

Show LaTeX source

Three particles $A$, $B$ and $C$ of masses 0.3 kg, 0.4 kg and $m$ kg respectively lie at rest in a straight line on a smooth horizontal plane. The distance between $B$ and $C$ is 2.1 m. $A$ is projected directly towards $B$ with speed $2 \text{ m s}^{-1}$. After $A$ collides with $B$ the speed of $A$ is reduced to $0.6 \text{ m s}^{-1}$, still moving in the same direction.

\begin{enumerate}[label=(\alph*)]
\item Show that the speed of $B$ after the collision is $1.05 \text{ m s}^{-1}$. [2]
\end{enumerate}

After the collision between $A$ and $B$, $B$ moves directly towards $C$. Particle $B$ now collides with $C$. After this collision, the two particles coalesce and have a combined speed of $0.5 \text{ m s}^{-1}$.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find $m$. [2]
\end{enumerate}

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the time that it takes, from the instant when $B$ and $C$ collide, until $A$ collides with the combined particle. [5]
\end{enumerate}

\hfill \mbox{\textit{CAIE M1 2022 Q6 [9]}}

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