AQA AS Level Mathematics Paper 1, June 2025: Question 20

✅ Correct Answer

1.8 m s⁻¹

Units are not strictly penalised, but always write them to maintain good practice.

📐 Calculation

1. "Initial" means at time t = 0.
2. Substitute t = 0 into the speed model:
   v = 1.8 + 3.8(0) − 0.25(0)²
3. v = 1.8 m s⁻¹

🧠 Exam Technique

• The word "State" signals that no complex working is required—read the constant term directly from the formula.

❌ Common Errors

• Assuming initial speed must be 0 because a race begins from rest (do not confuse real life with the equation given!).

✅ Correct Answer

a = 3.8 − 0.5t

💡 Key Knowledge

Acceleration is the rate of change of velocity with respect to time:

a = dv/dt

• Differentiating a constant gives 0: d/dt(1.8) = 0
• Power rule: d/dt(c·tⁿ) = n·c·tⁿ⁻¹

📐 Step-by-Step Differentiation

1. Given: v = 1.8 + 3.8t − 0.25t²
2. Differentiate term-by-term with respect to t:
   • d/dt(1.8) = 0
   • d/dt(3.8t) = 3.8
   • d/dt(−0.25t²) = −2 × 0.25t = −0.5t
3. Combine terms: a = 3.8 − 0.5t

❌ Common Errors

• Sign errors: writing +0.5t instead of −0.5t.
• Attempting to use constant acceleration (SUVAT) formulas—these cannot be used because acceleration depends on time.

✅ Correct Answer

16.2 m s⁻¹ (or 16.24 m s⁻¹)

🧠 Exam Technique: "Hence" & "Fully Justify"

• "Hence" means you must use your acceleration from part (b)(i).
• A maximum speed occurs when the gradient dv/dt = 0 (acceleration is zero).
• To fully justify, show the derivation of t from setting a = 0, or check the second derivative: d²v/dt² = −0.5 < 0 (confirming a local maximum).

📐 Step-by-Step Calculation

1. Set acceleration to zero:
   3.8 − 0.5t = 0
2. Solve for t:
   0.5t = 3.8 ⇒ t = 3.8 / 0.5 = 7.6 s
3. Substitute t = 7.6 into the original velocity formula:
   v_max = 1.8 + 3.8(7.6) − 0.25(7.6)²
   v_max = 1.8 + 28.88 − 14.44
   v_max = 16.24 m s⁻¹ (or 16.2 m s⁻¹ to 3 s.f.)

❌ Common Errors

• Stopping after finding t = 7.6 s without calculating the speed.
• Substituting t = 7.6 back into the acceleration formula instead of the velocity formula.
• Arithmetic slips when squaring 7.6 or handling the signs.

✅ Acceptable Model Evaluations

• Comparison of Max Speed: The model gives 16.2 m s⁻¹, which is significantly greater than the recorded 12.4 m s⁻¹ (16.2 > 12.4). Therefore, the model seriously overestimates speed and is not accurate.
• Initial Condition Flaw: According to the model, the athlete's initial speed is 1.8 m s⁻¹, whereas a sprinter starts from rest (0 m s⁻¹). Hence, the model is not realistic.

🧠 Exam Technique

• Evaluation questions require both a numerical comparison and a clear conclusion.
• Never just write numbers down. Explicitly state: "...therefore the model is not accurate".

❌ Common Errors

• Writing vague statements such as "it is close" or "it is quite accurate" without directly comparing the values.
• Failing to provide a clear concluding verdict on the accuracy.