CAIE Further Paper 3 2022 June — Question 7 11 marks

Exam BoardCAIE
ModuleFurther Paper 3 (Further Paper 3)
Year2022
SessionJune
Marks11
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicProjectiles
TypeTwo projectiles meeting - 2D flight
DifficultyChallenging +1.2 This is a standard two-particle projectile collision problem requiring systematic application of kinematic equations. While it involves multiple particles with different launch times and angles, the solution follows a clear algorithmic approach: resolve velocities using given tan values, equate horizontal displacements, solve the resulting equation, then find collision coordinates. The algebra is moderately involved but straightforward for Further Maths students.
Spec3.02d Constant acceleration: SUVAT formulae3.02i Projectile motion: constant acceleration model
Particles \(P\) and \(Q\) are projected in the same vertical plane from a point \(O\) at the top of a cliff. The height of the cliff exceeds 50 m. Both particles move freely under gravity. Particle \(P\) is projected with speed \(\frac{35}{2} \text{ m s}^{-1}\) at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac{4}{3}\). Particle \(Q\) is projected with speed \(u \text{ m s}^{-1}\) at an angle \(\beta\) above the horizontal, where \(\tan \beta = \frac{1}{2}\). Particle \(Q\) is projected one second after the projection of particle \(P\). The particles collide \(T\) s after the projection of particle \(Q\).
  1. Write down expressions, in terms of \(T\), for the horizontal displacements of \(P\) and \(Q\) from \(O\) when they collide and hence show that \(4uT = 21\sqrt{5(T + 1)}\). [4]
  2. Find the value of \(T\). [4]
  3. Find the horizontal and vertical displacements of the particles from \(O\) when they collide. [3]
Question 7:
AnswerMarks Guidance
7(a)For Q: xucosT B1
35
For P: x cosT 1
AnswerMarks
2B1
35
Collision, so cosT 1ucosT
AnswerMarks Guidance
2M1 Equate and attempt to rearrange.
35 3 2
 T 1u T
2 5 5
AnswerMarks Guidance
4uT 21 5T 1A1 AG
Shown convincingly.
4
AnswerMarks
7(b)Vertical motion to collision:
1
For Q: yusinT  gT2
2
35 1
For P: y sinT 1 g(T 1)2
AnswerMarks Guidance
2 2M1 A1 M1 for both expressions, one correct.
1 1 35 4 1
Equate: u T  gT2   T 1 g(T 1)2
5 2 2 5 2
14T 1 1 g  T2 2T 1T2  21 T 1
AnswerMarks Guidance
2 4M1 Equate and attempt to solve
16T 3621T 21, 1 55T
AnswerMarks
T 3A1
4
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks Guidance
7(c)x42 B1
y 24M1
y24 (or 24 m below O)A1 Correct sign or in words.
3
Question 7:
--- 7(a) ---
7(a) | For Q: xucosT | B1
35
For P: x cosT 1
2 | B1
35
Collision, so cosT 1ucosT
2 | M1 | Equate and attempt to rearrange.
35 3 2
 T 1u T
2 5 5
4uT 21 5T 1 | A1 | AG
Shown convincingly.
4
--- 7(b) ---
7(b) | Vertical motion to collision:
1
For Q: yusinT  gT2
2
35 1
For P: y sinT 1 g(T 1)2
2 2 | M1 A1 | M1 for both expressions, one correct.
1 1 35 4 1
Equate: u T  gT2   T 1 g(T 1)2
5 2 2 5 2
14T 1 1 g  T2 2T 1T2  21 T 1
2 4 | M1 | Equate and attempt to solve
16T 3621T 21, 1 55T
T 3 | A1
4
Question | Answer | Marks | Guidance
--- 7(c) ---
7(c) | x42 | B1
y 24 | M1
y24 (or 24 m below O) | A1 | Correct sign or in words.
3
Particles $P$ and $Q$ are projected in the same vertical plane from a point $O$ at the top of a cliff. The height of the cliff exceeds 50 m. Both particles move freely under gravity. Particle $P$ is projected with speed $\frac{35}{2} \text{ m s}^{-1}$ at an angle $\alpha$ above the horizontal, where $\tan \alpha = \frac{4}{3}$. Particle $Q$ is projected with speed $u \text{ m s}^{-1}$ at an angle $\beta$ above the horizontal, where $\tan \beta = \frac{1}{2}$. Particle $Q$ is projected one second after the projection of particle $P$. The particles collide $T$ s after the projection of particle $Q$.

\begin{enumerate}[label=(\alph*)]
\item Write down expressions, in terms of $T$, for the horizontal displacements of $P$ and $Q$ from $O$ when they collide and hence show that $4uT = 21\sqrt{5(T + 1)}$. [4]

\item Find the value of $T$. [4]

\item Find the horizontal and vertical displacements of the particles from $O$ when they collide. [3]
\end{enumerate}

\hfill \mbox{\textit{CAIE Further Paper 3 2022 Q7 [11]}}
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