CAIE Further Paper 3 2023 June — Question 7 9 marks

Exam BoardCAIE
ModuleFurther Paper 3 (Further Paper 3)
Year2023
SessionJune
Marks9
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicProjectiles
TypeFinding angle given constraints
DifficultyStandard +0.8 This is a multi-step projectile motion problem requiring derivation of standard formulae, then solving a non-trivial constraint equation involving the descent distance. Part (b) requires setting up and solving a quadratic equation linking the angle to the height condition, which goes beyond routine projectile calculations. The problem-solving aspect and algebraic manipulation elevate this above average difficulty.
Spec3.02d Constant acceleration: SUVAT formulae3.02i Projectile motion: constant acceleration model
At time \(t\)s, a particle \(P\) is projected with speed \(40\)m s\(^{-1}\) at an angle \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The greatest height achieved by \(P\) during its flight is \(H\)m and the corresponding time is \(T\)s.
  1. Obtain expressions for \(H\) and \(T\) in terms of \(\theta\). [2]
During the time between \(t = T\) and \(t = 3\), \(P\) descends a distance \(\frac{1}{4}H\).
  1. Find the value of \(\theta\). [4]
  2. Find the speed of \(P\) when \(t = 3\). [3]
Question 7:
AnswerMarks
7(a)800sin2
H 80sin2 or
AnswerMarks
gB1
40sin
T 4sin or
AnswerMarks
gB1
2
AnswerMarks
7(b)1 1
103T2
Between t T and t 3  H 
AnswerMarks Guidance
4 2M1 A1 No extra terms.
1
80sin2534sin2
Use results from part (a)
4
AnswerMarks Guidance
4sin28sin30M1 Substitute their expressions for H and T from
part (a) and obtain a quadratic equation in sin
with no more than three terms.
1
sin , 30
AnswerMarks Guidance
2A1 Single answer. NFWW.
Alternative method for question 7 part (b)
3 1
H  y3403 sin 1032
AnswerMarks Guidance
4 2M1 A1 120 sin45
3
Use results from (a): 80sin2120 sin45
4
AnswerMarks Guidance
4sin28sin30M1 Substitute their expressions for H and T from
part (a) and obtain a quadratic equation in sin
with no more than three terms.
1
sin , 30
AnswerMarks Guidance
2A1 Single answer. NFWW.
4
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks Guidance
7(c)When t 3 speeds 40cos and 40sin103 B1
Square and add to find square of speed: v2   20 3 2 102M1 Must be numerical.
v2 1300, v10 13 [= 36.1]A1
3
Question 7:
--- 7(a) ---
7(a) | 800sin2
H 80sin2 or
g | B1
40sin
T 4sin or
g | B1
2
--- 7(b) ---
7(b) | 1 1
103T2
Between t T and t 3  H 
4 2 | M1 A1 | No extra terms.
1
80sin2534sin2
Use results from part (a)
4
4sin28sin30 | M1 | Substitute their expressions for H and T from
part (a) and obtain a quadratic equation in sin
with no more than three terms.
1
sin , 30
2 | A1 | Single answer. NFWW.
Alternative method for question 7 part (b)
3 1
H  y3403 sin 1032
4 2 | M1 A1 | 120 sin45
3
Use results from (a): 80sin2120 sin45
4
4sin28sin30 | M1 | Substitute their expressions for H and T from
part (a) and obtain a quadratic equation in sin
with no more than three terms.
1
sin , 30
2 | A1 | Single answer. NFWW.
4
Question | Answer | Marks | Guidance
--- 7(c) ---
7(c) | When t 3 speeds 40cos and 40sin103 | B1
Square and add to find square of speed: v2   20 3 2 102 | M1 | Must be numerical.
v2 1300, v10 13 [= 36.1] | A1
3
At time $t$s, a particle $P$ is projected with speed $40$m s$^{-1}$ at an angle $\theta$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. The greatest height achieved by $P$ during its flight is $H$m and the corresponding time is $T$s.

\begin{enumerate}[label=(\alph*)]
\item Obtain expressions for $H$ and $T$ in terms of $\theta$. [2]
\end{enumerate}

During the time between $t = T$ and $t = 3$, $P$ descends a distance $\frac{1}{4}H$.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the value of $\theta$. [4]

\item Find the speed of $P$ when $t = 3$. [3]
\end{enumerate}

\hfill \mbox{\textit{CAIE Further Paper 3 2023 Q7 [9]}}
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Q1 5 Q2 5 Q3 7 Q4 8 Q5 7 Q6 9 Q7 9