CAIE M1 2022 June — Question 4 9 marks

Exam BoardCAIE
ModuleM1 (Mechanics 1)
Year2022
SessionJune
Marks9
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicTravel graphs
TypeTwo-particle meeting or overtaking
DifficultyStandard +0.3 This is a standard mechanics kinematics problem requiring application of SUVAT equations and interpretation of motion. Part (a) involves straightforward use of s=ut+½at² for constant acceleration motion. Part (b) requires solving a quadratic equation. Part (c) tests graph sketching skills. While it has multiple parts and requires careful attention to the 4-second delay, all techniques are routine for M1 students with no novel problem-solving insight needed. Slightly easier than average due to being a textbook-style multi-part question.
Spec3.02b Kinematic graphs: displacement-time and velocity-time3.02d Constant acceleration: SUVAT formulae
A particle \(A\), moving along a straight horizontal track with constant speed \(8\text{ms}^{-1}\), passes a fixed point \(O\). Four seconds later, another particle \(B\) passes \(O\), moving along a parallel track in the same direction as \(A\). Particle \(B\) has speed \(20\text{ms}^{-1}\) when it passes \(O\) and has a constant deceleration of \(2\text{ms}^{-2}\). \(B\) comes to rest when it returns to \(O\).
  1. Find expressions, in terms of \(t\), for the displacement from \(O\) of each particle \(t\) seconds after \(B\) passes \(O\). [3]
  2. Find the values of \(t\) when the particles are the same distance from \(O\). [3]
  3. On the given axes, sketch the displacement-time graphs for both particles, for values of \(t\) from \(0\) to \(20\). [3] $$s \text{ (m)}$$ $$200$$ $$100$$ $$0 \quad 0 \quad 10 \quad 20 \quad t \text{ (s)}$$
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 Guidance
4(a)Use suvat to find expressions for s or s
A BM1 For s must be using u8 and time of t4
A
1
For s , using s ut at2 with u 20 and a2
B 2
s 84t 328t
AnswerMarks Guidance
AA1 Any unsimplified expression; ISW
1
s 20t 2t2
AnswerMarks Guidance
B 2A1 Any unsimplified expression; ISW
If 0 marks scored then allow
SC: B1 for s 8t and
A
1
s 20t4 2t42
B1 for maximum 2/3
B 2
3
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks
4(b)1
84t20t 2t2
AnswerMarks Guidance
2*M1 Equating their expressions for s and s to form an equation
A B
in t
where s is of the form 8t32
A
1
and s is of the form 20t 2t2
B 2
AnswerMarks Guidance
Attempt to solve a 3-term quadratic to find at least one t valueDM1 For reference t2 12t320
Allow if no working seen and have correct real solution(s) to
their 3-term quadratic.
If working shown and if using the formula, it must be using
the correct formula. If factorising must have 3 of the 4 terms
correct of t4t8
AnswerMarks Guidance
t 4 and 8A1 If 0 marks scored then allow
1
SC: M1 for 8t 20t4 2t42 and
2
A1 for t 8 and 12 maximum 2/3.
3
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks Guidance
4(c)Straight line B1 FT
Full domain not required.
FT if they get s 8t using the SC in (a)
A
AnswerMarks Guidance
Inverted quadratic, passing through origin.B1 FT Full domain not required but must clearly go beyond the
maximum.
1
FT if they get s 20t4 2t42 using the SC
B 2
in (a), with curve though positive t axis before turning point.
All correct, line through (0, 32), quadratic through (20, 0), intersections
AnswerMarks Guidance
indicated at t 4 and t 8.B1 Intersections must occur before the turning point.
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) | Use suvat to find expressions for s or s
A B | M1 | For s must be using u8 and time of t4
A
1
For s , using s ut at2 with u 20 and a2
B 2
s 84t 328t
A | A1 | Any unsimplified expression; ISW
1
s 20t 2t2
B 2 | A1 | Any unsimplified expression; ISW
If 0 marks scored then allow
SC: B1 for s 8t and
A
1
s 20t4 2t42
B1 for maximum 2/3
B 2
3
Question | Answer | Marks | Guidance
--- 4(b) ---
4(b) | 1
84t20t 2t2
2 | *M1 | Equating their expressions for s and s to form an equation
A B
in t
where s is of the form 8t32
A
1
and s is of the form 20t 2t2
B 2
Attempt to solve a 3-term quadratic to find at least one t value | DM1 | For reference t2 12t320
Allow if no working seen and have correct real solution(s) to
their 3-term quadratic.
If working shown and if using the formula, it must be using
the correct formula. If factorising must have 3 of the 4 terms
correct of t4t8
t 4 and 8 | A1 | If 0 marks scored then allow
1
SC: M1 for 8t 20t4 2t42 and
2
A1 for t 8 and 12 maximum 2/3.
3
Question | Answer | Marks | Guidance
--- 4(c) ---
4(c) | Straight line | B1 FT | Positive gradient, intersecting positive s axis.
Full domain not required.
FT if they get s 8t using the SC in (a)
A
Inverted quadratic, passing through origin. | B1 FT | Full domain not required but must clearly go beyond the
maximum.
1
FT if they get s 20t4 2t42 using the SC
B 2
in (a), with curve though positive t axis before turning point.
All correct, line through (0, 32), quadratic through (20, 0), intersections
indicated at t 4 and t 8. | B1 | Intersections must occur before the turning point.
3
Question | Answer | Marks | Guidance
A particle $A$, moving along a straight horizontal track with constant speed $8\text{ms}^{-1}$, passes a fixed point $O$. Four seconds later, another particle $B$ passes $O$, moving along a parallel track in the same direction as $A$. Particle $B$ has speed $20\text{ms}^{-1}$ when it passes $O$ and has a constant deceleration of $2\text{ms}^{-2}$. $B$ comes to rest when it returns to $O$.

\begin{enumerate}[label=(\alph*)]
\item Find expressions, in terms of $t$, for the displacement from $O$ of each particle $t$ seconds after $B$ passes $O$. [3]

\item Find the values of $t$ when the particles are the same distance from $O$. [3]

\item On the given axes, sketch the displacement-time graphs for both particles, for values of $t$ from $0$ to $20$. [3]

$$s \text{ (m)}$$
$$200$$

$$100$$

$$0 \quad 0 \quad 10 \quad 20 \quad t \text{ (s)}$$
\end{enumerate}

\hfill \mbox{\textit{CAIE M1 2022 Q4 [9]}}
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