CAIE Further Paper 3 2024 June — Question 7 11 marks

Exam BoardCAIE
ModuleFurther Paper 3 (Further Paper 3)
Year2024
SessionJune
Marks11
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TopicVariable acceleration (1D)
TypeAcceleration as function of velocity (separation of variables)
DifficultyStandard +0.8 This is a standard variable acceleration problem requiring setting up and solving a differential equation (F=ma with resistance proportional to v), then integrating to find displacement. While it involves multiple steps and integration techniques, the method is well-practiced in Further Maths mechanics. The resistance being mv (not mv²) makes the differential equation separable and straightforward. More challenging than typical A-level due to the multi-step nature and being Further Maths content, but follows a standard template without requiring novel insight.
Spec4.10c Integrating factor: first order equations6.02a Work done: concept and definition
A parachutist of mass \(m\) kg opens his parachute when he is moving vertically downwards with a speed of \(50\text{ ms}^{-1}\). At time \(t\) s after opening his parachute, he has fallen a distance \(x\) m from the point where he opened his parachute, and his speed is \(v\text{ ms}^{-1}\). The forces acting on him are his weight and a resistive force of magnitude \(mv\) N.
  1. Find an expression for \(v\) in terms of \(t\). [6]
  2. Find an expression for \(x\) in terms of \(t\). [3]
  3. Find the distance that the parachutist has fallen, since opening his parachute, when his speed is \(15\text{ ms}^{-1}\). [2]
Question 7:
AnswerMarks
7(a)dv
m m10v
AnswerMarks Guidance
dtB1 No marks in this part if suvat used.
Must have sight of m (for example in F = ma).
AnswerMarks Guidance
ln10v t A or lnv10t A*M1 A1 Separate variables and integrate to obtain a ln
term. Constant may be omitted.
Constant needed for A1
AnswerMarks Guidance
Use t 0, v50: Aln 40DM1 Find constant, dependent on previous M1.
May use limits instead.
AnswerMarks Guidance
0.1vet 4etM1 Remove all logs .
v1040etA1 Correct work only .
6
AnswerMarks Guidance
7(b)x10t40et B M1
part (b).
Integrate their answer to part (a).
Constant may be omitted.
AnswerMarks Guidance
Use t 0, x0: B40M1 Use initial condition in their expression for x in
terms of t .
AnswerMarks
x10t40et 40A1
3
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks
7(c)1
When v15, et  , t 2.08 or ln 8
AnswerMarks Guidance
8M1 No marks in this part if suvat used in part (a) part
(b) or part (c).
Find value of t from their answer to part (a).
AnswerMarks Guidance
x 55.8 (metres)A1 Note 35 + 10 ln 8 scores A0.
2
Question 7:
--- 7(a) ---
7(a) | dv
m m10v
dt | B1 | No marks in this part if suvat used.
Must have sight of m (for example in F = ma).
ln10v t A or lnv10t A | *M1 A1 | Separate variables and integrate to obtain a ln
term. Constant may be omitted.
Constant needed for A1
Use t 0, v50: Aln 40 | DM1 | Find constant, dependent on previous M1.
May use limits instead.
0.1vet 4et | M1 | Remove all logs .
v1040et | A1 | Correct work only .
6
--- 7(b) ---
7(b) | x10t40et B | M1 | No marks in this part if suvat used in part (a) or
part (b).
Integrate their answer to part (a).
Constant may be omitted.
Use t 0, x0: B40 | M1 | Use initial condition in their expression for x in
terms of t .
x10t40et 40 | A1
3
Question | Answer | Marks | Guidance
--- 7(c) ---
7(c) | 1
When v15, et  , t 2.08 or ln 8
8 | M1 | No marks in this part if suvat used in part (a) part
(b) or part (c).
Find value of t from their answer to part (a).
x 55.8 (metres) | A1 | Note 35 + 10 ln 8 scores A0.
2
A parachutist of mass $m$ kg opens his parachute when he is moving vertically downwards with a speed of $50\text{ ms}^{-1}$. At time $t$ s after opening his parachute, he has fallen a distance $x$ m from the point where he opened his parachute, and his speed is $v\text{ ms}^{-1}$. The forces acting on him are his weight and a resistive force of magnitude $mv$ N.

\begin{enumerate}[label=(\alph*)]
\item Find an expression for $v$ in terms of $t$. [6]

\item Find an expression for $x$ in terms of $t$. [3]

\item Find the distance that the parachutist has fallen, since opening his parachute, when his speed is $15\text{ ms}^{-1}$. [2]
\end{enumerate}

\hfill \mbox{\textit{CAIE Further Paper 3 2024 Q7 [11]}}
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Q1 4 Q2 6 Q3 7 Q4 7 Q5 7 Q6 8 Q7 11