## ERQ · 10 marks · Topics: C.5 Doppler effect + A.1 Kinematics · Archetype: theory_application
**Integration:** primary=C.5 Doppler effect, secondary=A.1 Kinematics (strength: supporting)
**Stem.** An ambulance drives along a long, straight road at a constant speed of 25 m s⁻¹ toward a pedestrian standing on the pavement. The ambulance siren emits sound of frequency 850 Hz in the reference frame of the ambulance. The speed of sound in still air on this day is 340 m s⁻¹. The pedestrian records the apparent frequency of the siren continuously: first as the ambulance approaches, then as it passes a point closest to the pedestrian (the perpendicular distance from the pedestrian to the road is small but non-zero), and finally as it recedes. In a separate trial on the same road, the ambulance instead accelerates uniformly from rest over a distance of 80 m before passing the pedestrian.
### Part (a) State [2 marks] · AO1 · Topic: C.5
State two assumptions of the Doppler effect equation `f' = f (v / (v − u_s))` as given in the data booklet when applied to the approaching ambulance.
### Part (b) Calculate [3 marks] · AO2 · Topic: C.5
Calculate the apparent frequency heard by the pedestrian while the ambulance is approaching at constant 25 m s⁻¹.
### Part (c) Determine [3 marks] · AO2 · Topic: C.5
Determine the difference between the apparent frequency heard while the ambulance is approaching at 25 m s⁻¹ and the apparent frequency heard while it is receding at the same speed.
### Part (d) Explain [2 marks] · AO3 · Topic: C.5+A.1 · ASSUMPTIONS DISCRIMINATOR
In the second trial the ambulance accelerates uniformly from rest, reaching the pedestrian after travelling 80 m. Using ideas from kinematics, explain why the apparent frequency recorded during this trial cannot be predicted by a single application of the Doppler equation.
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## Mark Scheme
### Part (a) [2 marks]
- M1: source moves directly along the line joining source and observer / observer is stationary relative to the medium (air) [no ECF]
- M2: source speed is constant AND less than the speed of sound in the medium / medium (air) is uniform and stationary [no ECF]
### Part (b) [3 marks] — Calculate
- M1: correct substitution into f' = f · v / (v − u_s) = 850 × 340 / (340 − 25) [no ECF]
- M2: correct evaluation of denominator 315 (m s⁻¹) or ratio 340/315 ≈ 1.0794 [no ECF]
- M3: f' ≈ 917 Hz (accept 917–918 Hz) [ECF from M1]
### Part (c) [3 marks] — Determine
- M1: applies receding form f'' = f · v / (v + u_s) = 850 × 340 / (340 + 25) [no ECF]
- M2: f'' ≈ 792 Hz (accept 791–792 Hz) [ECF from M1]
- M3: Δf = f' − f'' ≈ 917 − 792 ≈ 125 Hz (accept 125–126 Hz) [ECF from (b) and M2]
### Part (d) [2 marks] — Explain (causal chain per §4.4.1)
- M1: from kinematics, uniform acceleration over 80 m from rest means the source speed u_s changes continuously with time (e.g. v² = u² + 2as gives a final speed of ~ a·t that is not constant) [fact / observation]
- M2: therefore the Doppler factor v/(v − u_s) changes continuously, so the apparent frequency rises with time as the ambulance approaches and a single substitution with one value of u_s cannot capture this — the equation requires constant source speed [therefore / consequence linking to C.5 assumption]
### Marker notes
- Part (a): accept also "wavelength in front of source is uniformly compressed" or "no wind / air at rest"; do NOT award for restating the equation.
- Part (b) alternative: wavelength method λ' = (v − u_s)/f = 315/850 = 0.3706 m, then f' = v/λ' = 340/0.3706 ≈ 917 Hz.
- Part (c) accept rounding range 124–126 Hz; full ECF if (b) used a wrong but consistent value.
- Part (d) accept any of: source speed not constant / Doppler equation assumes constant u_s / apparent frequency would itself vary with time during the approach / would need calculus or instantaneous u_s at each moment.