Challenging +1.2 This is a standard projectile motion problem requiring substitution of two conditions into the trajectory equation to form simultaneous equations. While it involves algebraic manipulation and solving for two unknowns, the approach is methodical and the angle tan⁻¹(2) gives clean components. The 5-mark allocation and straightforward setup place it moderately above average difficulty but well within typical A-level mechanics scope.
A particle \(P\) is projected with speed \(u \text{ m s}^{-1}\) at an angle \(\tan^{-1} 2\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. When \(P\) has travelled a distance \(56 \text{ m}\) horizontally from \(O\), it is at a vertical height \(H \text{ m}\) above the plane. When \(P\) has travelled a distance \(84 \text{ m}\) horizontally from \(O\), it is at a vertical height \(\frac{1}{2}H \text{ m}\) above the plane.
Find, in either order, the value of \(u\) and the value of \(H\). [5]
Use equation of trajectory with point (56, H) or 84, H
2
5g 1 5g
H =112− 562 or H =168− 842
Answer
Marks
Guidance
2u2 2 2u2
M1
For one equation with one error.
A1
Both correct.
Eliminate to find u or H
M1
u=35
A1
H = 48
A1
5
Answer
Marks
Guidance
Question
Answer
Marks
Question 1:
1 | 1
Use equation of trajectory with point (56, H) or 84, H
2
5g 1 5g
H =112− 562 or H =168− 842
2u2 2 2u2 | M1 | For one equation with one error.
A1 | Both correct.
Eliminate to find u or H | M1
u=35 | A1
H = 48 | A1
5
Question | Answer | Marks | Guidance
A particle $P$ is projected with speed $u \text{ m s}^{-1}$ at an angle $\tan^{-1} 2$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. When $P$ has travelled a distance $56 \text{ m}$ horizontally from $O$, it is at a vertical height $H \text{ m}$ above the plane. When $P$ has travelled a distance $84 \text{ m}$ horizontally from $O$, it is at a vertical height $\frac{1}{2}H \text{ m}$ above the plane.
Find, in either order, the value of $u$ and the value of $H$. [5]
\hfill \mbox{\textit{CAIE Further Paper 3 2024 Q1 [5]}}