AQA A-Level Mathematics Paper 2, June 2025: Question 13

3 marks · Easy difficulty · Modelling

Use constant acceleration equations to show a car starts from rest, and explain why the constant acceleration model would eventually become invalid.

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Question

Question 13 consists of two parts. Context states: A car moves in a straight line on a horizontal road from point A to point B. Distance AB is 225 metres. At point B, the velocity of the car is 30 m s⁻¹. The car is modelled with a constant acceleration of 2 m s⁻². Part (a) asks to show that the car starts from rest at point A, for 2 marks. Part (b) states the car continues past point B in the same straight line and asks to explain why the model would eventually become invalid, for 1 mark.
Question text

13 A car moves, in a straight line on a horizontal road, from a point A to a point B

The distance AB is 225 metres.

At the point B, the velocity of the car is 30 m s–1

The car is modelled as having a constant acceleration of 2 m s–2

13 (a) Show that the car starts from rest at point A

[2 marks]

13 (b) The car continues to move past the point B in the same straight line.

Explain why the model would eventually become invalid.

[1 mark]

Mark scheme

Show the mark scheme Mark scheme for Question 13. For part (a), M1 is awarded for substituting a = 2, s = 225, v = 30 and u into v² = u² + 2as. R1 is awarded for completing a reasoned argument with at least one intermediate step (e.g. 30² = u² + 2(2)(225) leads to 900 = u² + 900, giving u² = 0, so u = 0 m s⁻¹) to conclude that the car is at rest at point A. For part (b), E1 is awarded for explaining why the model becomes invalid, such as: acceleration is unlikely to remain constant, the car reaches its maximum speed, the road will not remain straight/obstacles, or the car runs out of fuel.

Q Marking instructions AO Marks Typical solution

Substitutes a = 2 s = 225, v = 30 3.3 M1 a = 2 s = 225, v = 30

13(a) and u into v2 = u2 + 2as OE v2 = u2 + 2as

Completes reasoned argument, 2.1 R1 302 = u2 + 2 2 225

with at least one intermediate 2

step after the initial substitution, 900 = u + 900

to obtain u = 0 or u2 = 0 2

u = 0

and −1

concludes the car starts from u = 0m s

rest Therefore, the car is at rest at

AG

Accept making u the subject of point A.

v2 = u2 + 2as before substitution

as the intermediate step

Subtotal 2

13(b) Explains why the model 3.5b E1 A car’s acceleration is unlikely to stay

becomes invalid, eg: at the same rate.

• That acceleration is unlikely to

remain constant

Or

gives a contextual reason as to

why the car may have to slow

down or stop, eg:

• That the car reaches its

maximum speed

• The road will not remain

straight or there will be an

obstacle

• The car will run out of fuel

Subtotal 1

Question 13 Total 3

How to answer it

Constant Acceleration & Modelling Validity (Question 13)

📌 What this question tests

Core syllabus skills assessed:

  • Selecting and correctly applying standard constant acceleration (suvat) formulae in 1D horizontal motion.
  • Structuring a rigorous mathematical proof/reasoning step ("Show that...") concluding with clear mathematical deduction.
  • Critically evaluating mathematical models in mechanics and providing realistic contextual limitations.
Question 13 (a) • 2 Marks

Show that the car starts from rest at point A

AO3.3 (Modelling) & AO2.1 (Reasoning)

💡 Key Knowledge

  • Kinematic variables given:
    • Displacement, s = 225 m
    • Final velocity, v = 30 m s⁻¹
    • Acceleration, a = 2 m s⁻²
    • Initial velocity, u = ?
  • Formula selection: Choose the formula linking s , v , a , and u (no time t ):
    v² = u² + 2as
  • "Starts from rest" means initial speed u = 0 m s⁻¹ .

📐 Step-by-Step Proof

  1. State equation & substitute values:
    v² = u² + 2as
    30² = u² + 2(2)(225)
  2. Simplify numerical terms:
    900 = u² + 900
  3. Solve for u:
    u² = 0 ⇒ u = 0 m s⁻¹
  4. Conclude clearly:
    Since u = 0 , the car starts from rest at point A.

✅ Mark Scheme Breakdown

  • [M1] (AO3.3): Correct substitution of a = 2 , s = 225 , and v = 30 into v² = u² + 2as (or equivalent rearrangement with u as subject).
  • [R1] (AO2.1): Fully completed reasoned argument showing at least one clear intermediate step after substitution (e.g. 900 = u² + 900 or u² = 0 ) and an explicit conclusion that the car is at rest at point A.

❌ Common Errors & Pitfalls

  • Circular reasoning (Assuming the conclusion): Starting by assuming u = 0 and calculating v = √(2 × 2 × 225) = 30 . While mathematically valid, this must be concluded carefully; the safest and expected method is finding u directly.
  • Missing intermediate steps: Jumping straight from 30² = u² + 2(2)(225) to u = 0 without showing u² = 900 - 900 = 0 loses the reasoning mark R1 in an "AG" (Answer Given) question.
  • Forgetting the conclusion: Leaving the answer at u = 0 without writing a statement like "Therefore, car starts from rest".
Question 13 (b) • 1 Mark

Explain why the model would eventually become invalid

AO3.5b (Evaluation)

🧠 Exam Technique: Modelling Limitations

To score this evaluation mark, identify the key assumption in the model and explain why it cannot hold indefinitely in the real world.

  • Core assumption: The acceleration is constant at 2 m s⁻² forever in a straight line.
  • Give either a mechanical/kinematic reason (e.g., speed, engine limits) or a practical contextual reason (e.g., fuel, road geometry).

✅ Acceptable Answers [E1]

Any one of the following valid explanations earns the mark:

  • Acceleration cannot remain constant: A car cannot maintain constant acceleration indefinitely due to air resistance / drag increasing with speed.
  • Speed limit / Maximum speed: The car will eventually reach its maximum possible velocity (top speed).
  • Road limitations: The road will not continue in a straight line forever, or there may be obstacles/traffic lights/junctions.
  • Fuel limitations: The car will eventually run out of fuel.

❌ Common Errors

  • Vague statements: Writing simply "cars can't drive forever" or "it's unrealistic" without specifying the mechanical or physical reason.
  • Ignoring the prompt: Proposing changes to the starting conditions rather than looking at what happens after point B as time increases.

Topics

Mechanics · Q: Kinematics

Question and mark scheme from the AQA A-Level Mathematics examination, Paper 2, June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.