AQA A-Level Physics Paper 1, June 2023: Question 30

1 mark · Medium difficulty · Multiple Choice

Determine the expression for tan theta for an aircraft of mass m travelling at constant speed v in a horizontal circular path of radius r while banking at an angle theta.

Practise this question

Question

A multiple-choice question showing an aircraft banking at an angle theta in a horizontal circular path. A diagram illustrates the aircraft tilted at angle theta with a vertical dashed line, showing a lift force acting at angle theta and weight mg acting downwards, and a radius r to the centre of the circular path. Four options A, B, C, D are provided as algebraic fractions containing v, r, and g.
Question text

30 When an aircraft turns in a horizontal circular path, it banks at an angle θ.

The aircraft has mass m and travels at constant speed v in a horizontal circular path of

radius r. The lift force acts at the angle θ.

What is tan θ?

[1 mark]

gv2

A

r

rv2

B

g

rg

C 2

v

v2

D

rg

Mark scheme

Show the mark scheme The mark scheme table shows question 30 with the correct answer D, corresponding to the expression v squared over r g.

v2

30 D

rg

How to answer it

Banked Aircraft in Circular Motion

What this question tests

This question assesses your ability to resolve forces in two dimensions, apply Newton's second law to circular motion, and derive algebraic expressions involving trigonometric functions for a banking vehicle or aircraft.

Question 30

Deriving the Angle of Bank (tan θ)

✅ Correct Answer

Option D: v² / (rg)

Worth 1 mark

💡 Key Knowledge

  • Vertical equilibrium: L cos θ = mg
  • Horizontal centripetal force: L sin θ = mv² / r
  • Trigonometric identity: sin θ / cos θ = tan θ

🧠 Exam Technique

For multiple-choice derivation questions, set up your vertical and horizontal force balance equations immediately. Divide the horizontal equation by the vertical equation to eliminate mass ( m ) and isolate tan θ .

❌ Common Errors

Students often mix up sine and cosine components relative to the given angle θ . Always check the diagram: if θ is measured from the vertical axis, the vertical component uses cos θ and the horizontal component uses sin θ .

📐 Step-by-Step Derivation

  1. Step 1 (Vertical Resolution): The upward vertical component of the lift force balances the downward weight of the aircraft.
    L cos θ = mg
  2. Step 2 (Horizontal Resolution): The horizontal component of the lift force provides the centripetal force required for circular motion.
    L sin θ =mv² / r
  3. Step 3 (Combine Equations): Divide the horizontal equation by the vertical equation to cancel out lift ( L ) and mass ( m ):
    (L sin θ) / (L cos θ) = (mv² / r) / (mg)
  4. Step 4 (Simplify): Simplify the fractions to arrive at the final result:
    tan θ = v² / (rg)

Topics

Physics · 3.6 Further mechanics and thermal physics (A-level only)

Question and mark scheme from the AQA A-Level Physics examination, Paper 1, June 2023. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.