Standard +0.3 This is a standard projectile motion problem requiring derivation of the range formula and a simple inequality manipulation. The setup is straightforward, the mathematics involves basic kinematics (SUVAT) and the double angle identity sin 2ΞΈ = 2sin ΞΈ cos ΞΈ, which are core A-level mechanics techniques. While it requires multiple steps for 6 marks, it follows a well-practiced procedure with no novel insight needed, making it slightly easier than average.
A ball is projected forward from a fixed point, \(P\), on a horizontal surface with an initial speed \(u\text{ ms}^{-1}\), at an acute angle \(\theta\) above the horizontal.
The ball needs to first land at a point at least \(d\) metres away from \(P\).
You may assume the ball may be modelled as a particle and that air resistance may be ignored.
Show that
$$\sin 2\theta \geq \frac{dg}{u^2}$$
[6 marks]
Question 17:
17 | Models vertical motion using a
suitable constant acceleration
equation, to find the time of flight.
Condone consistent sine/ cosine
error | 3.3 | M1 | Vertically:
2
0 = π‘π‘Γπ’π’sinππβ0.5Γπ¨π¨Γπ‘π‘
2π’π’sinππ
π‘π‘ =
Horizontally: π¨π¨
=
π₯π₯ π’π’ π‘π‘cosππ
= ,
2π’π’sinππ
π₯π₯ π’π’ π¨π¨ cosππ
sin2ΞΈ=2sinΞΈcosΞΈ
using
2
π’π’ sin2ππ
π₯π₯ =
Since ππ
π₯π₯ β₯ ππ
2
π’π’ sin2ππ
β₯ ππ
ππ
πππ¨π¨
2
Obtains correct equation or
inequality for t
Must use g not numerical value | 1.1b | A1
Models horizontal displacement
Condone consistent sine/ cosine
error | 3.3 | M1
Obtains correct equation or
inequality to model horizontal
displacement | 1.1b | A1
Eliminates , where t is time of flight,
from their horizontal and vertical
models to oπ‘π‘btain an expression in
ΞΈ
terms of u, g and only. | 3.4 | M1
Completes reasoned argument to
obtain stated result. Must justify
stated inequality must have used
correct resolution of u throughout | 2.1 | R1
Total | 6 | sin2ππ β₯
π’π’
Q | Marking Instructions | AO | Marks | Typical Solution
A ball is projected forward from a fixed point, $P$, on a horizontal surface with an initial speed $u\text{ ms}^{-1}$, at an acute angle $\theta$ above the horizontal.
The ball needs to first land at a point at least $d$ metres away from $P$.
You may assume the ball may be modelled as a particle and that air resistance may be ignored.
Show that
$$\sin 2\theta \geq \frac{dg}{u^2}$$
[6 marks]
\hfill \mbox{\textit{AQA Paper 2 2020 Q17 [6]}}