Generated ERQ

✓ passed A.2 Forces and momentum × C.2 Wave model 12 marks SL 3 passes 114.03s $0.5971

## ERQ · 12 marks · Topics: A.2 Forces and momentum + C.2 Wave model **Stem.** A tennis ball launcher is mounted 4.0 m from a vertical concrete wall and fires balls horizontally at the wall. Each ball has mass 0.057 kg and leaves the launcher at a horizontal speed of 20 m·s⁻¹. The launcher fires balls at a constant rate of 12 balls per second. In Test 1 the wall is bare concrete and the balls rebound horizontally with speed 14 m·s⁻¹. In Test 2 the wall is coated with a thick layer of soft putty and the balls embed in the putty on impact. A microphone placed near the wall records the sound of each impact; an investigator analyses the recording to count the firing rate using the periodicity of the sound pulses. Assume horizontal flight, negligible air resistance, and that each ball is in contact with the wall for the same short time interval. ### Part (a) Define [2 marks] · AO1 · Topic: A.2 Define linear momentum and state the impulse–momentum theorem in equation form. ### Part (b)(i) Calculate [3 marks] · AO2 · Topic: A.2 Calculate the average horizontal force exerted by the wall on the stream of balls during Test 1 (elastic-like rebound). ### Part (b)(ii) Determine [2 marks] · AO2/AO3 · Topic: A.2 Determine the ratio of the average force on the wall in Test 1 to that in Test 2 (balls embed in putty). ### Part (c) Show that [3 marks] · AO2/AO3 · Topic: A.2 + C.2 The microphone signal in Test 2 consists of a regular train of short pulses, one per impact. Show that the fundamental frequency of this periodic pulse train lies within the audible range for humans, and identify the wavelength of the corresponding sound in air (take the speed of sound in air as 340 m·s⁻¹). ### Part (d) Suggest [2 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR The investigator finds that the force measured by a pressure sensor mounted on the wall in Test 1 is noticeably smaller than the value calculated in (b)(i). Suggest two reasons for this discrepancy. --- ## Mark Scheme ### Part (a) [2 marks] - M1: Momentum defined as product of mass and velocity (p = mv), identified as a vector [ECF: no] - M2: Impulse–momentum theorem stated as F·Δt = Δp (or F = Δp/Δt) [ECF: no] ### Part (b)(i) [3 marks] - M1: Recognises F = (rate of change of momentum) = n·m·(v_f − v_i)/1 s, with v_f and v_i opposite signs OR Δv = 20 − (−14) = 34 m·s⁻¹ [ECF: no] - M2: Correct substitution: F = 12 × 0.057 × 34 (= 12 × 0.057 × (20 + 14)) [ECF: yes] - M3: F ≈ 23 N, with unit and 2 sig figs [ECF: yes] ### Part (b)(ii) [2 marks] - M1: Test 2 force = 12 × 0.057 × 20 = 13.7 N (Δv = 20 m·s⁻¹ since balls stop) [ECF: yes from b(i) approach] - M2: Ratio F₁/F₂ = 23/13.7 ≈ 1.7 (accept 34/20 = 1.7) [ECF: yes] ### Part (c) [3 marks] - M1: Pulse-train fundamental frequency f = 12 Hz (one pulse per impact, 12 impacts per second) [ECF: no] - M2: Identifies 12 Hz lies below the standard audible range (20 Hz – 20 kHz), so the *fundamental* is sub-audible; however higher harmonics of the sharp pulse fall in audible range — OR if candidate computes harmonic content correctly and concludes audibility via harmonics — award. [Accept reasoning that the impact sound itself contains audible frequencies even though the repetition rate is 12 Hz.] [ECF: yes] - M3: λ = v/f = 340/12 ≈ 28 m for the 12 Hz fundamental (or λ for a stated audible harmonic, e.g. 340/240 ≈ 1.4 m for the 20th harmonic), unit and value correct [ECF: yes] ### Part (d) [2 marks] - M1: One valid reason, e.g. collision is not perfectly elastic — energy lost to deformation/heat/sound so rebound speed varies; or balls strike at slightly different points/angles so not all momentum is transferred horizontally to the sensor [ECF: no] - M2: A second distinct reason, e.g. contact time spreads the impulse and the sensor measures peak instantaneous pressure rather than time-averaged force; air resistance reduces incoming speed below 20 m·s⁻¹; the sensor area is smaller than the impact footprint so some impulse misses it [ECF: no] ### Marker notes - (b)(i): Common error — using Δv = 6 m·s⁻¹ (treating rebound as same-direction) earns M1 only if formula correct; final answer ≈ 4.1 N awarded M3 by ECF. - (c) discriminator: This part bridges A.2 (impact rate) and C.2 (wave model — frequency, wavelength, audible range). Accept either route: (i) fundamental 12 Hz is below 20 Hz so inaudible as a tone, or (ii) the impact transient is broadband and audible despite low repetition rate. Both demonstrate correct application of f and λ = v/f. - (d) discriminator: Accept any two of — inelastic energy loss, air drag reducing v₁, contact-time effects vs. peak-force measurement, ball spin/oblique impact, sensor calibration/area mismatch, variability in launch speed. Do NOT accept "human error" or vague "friction".

Critic verdicts

Iteration 0 — failed

ao_ladder pass

The ERQ correctly obeys the cognitive ladder. Part (a) is AO1 with Define command term (state knowledge). Parts (b)(i)–(b)(ii) are AO2/AO3 with Calculate/Determine (apply concepts to novel situation). Part (c) is AO2/AO3 with Show that (derive/apply relationships). Part (d) is AO3 with Suggest (evaluate assumptions against data, highest cognitive demand). The progression from recall→apply→synthesise is sound and appropriate for 12-mark SL.
cross_theme fail

Part (c) is labeled as blending A.2 + C.2, but the wave model (C.2) is used only as a name-check calculation. The question asks to 'show that the fundamental frequency of the detected pulse train is 12 Hz' — this is purely momentum/timing (A.2: balls fired at 12 per second). The wavelength calculation v = fλ is a standard wave formula applied to the *result* of the momentum analysis, not an integral part of solving the problem. C.2 wave properties do not genuinely contribute to understanding or resolving the force/momentum dynamics.

FIX: Restructure part (c) to require candidates to use wave concepts (e.g., Doppler shift, wave superposition, or resonance in the wall structure) to explain why the measured frequency differs from the firing rate, or how the sound wavelength constrains the sensor's detection method. Alternatively, move the wavelength calculation to a separate part and make (c) focus entirely on momentum, then create a genuinely integrated C.2 question (e.g., how reflected sound waves from the wall interfere, or how acoustic impedance affects force sensor calibration).
discriminator pass

Part (d) explicitly asks students to 'suggest two reasons why the measured force is lower than the calculated value, with reference to the assumptions of the model.' This directly requires critique of the model's assumptions (e.g. perfectly elastic collisions, no air drag, head-on impact). The mark scheme confirms that valid answers involve challenging key assumptions rather than performing further calculations.
rule_of_one pass

Each part's mark count aligns well with the number of distinct award criteria. Part (a): 2 marks, 2 bullets ✓. Part (b)(i): 3 marks, 3 bullets (concept/formula, substitution, numerical answer) ✓. Part (b)(ii): 2 marks, 2 bullets (concept, calculation) ✓. Part (c): 3 marks, 3 bullets (frequency identification, formula application, wavelength with units) ✓. Part (d): 2 marks, 2 bullets (two distinct reasoning criteria) ✓. All alignments are exact or within the ±1 tolerance.
topic pass

Part (c) genuinely integrates both topics: it requires understanding that the firing rate (12 balls/s) from A.2 momentum analysis directly sets the frequency of the acoustic pulse train (C.2 wave model), then applies the wave equation v = fλ to find wavelength. Both conceptual frameworks are essential, not superficial.

Iteration 1 — failed

ao_ladder fail

The cognitive ladder is broken. Part (b)(i) is labeled AO2/AO3 with Calculate, but part (b)(ii) is labeled AO2 with Determine—both middle sections should progress to AO3. More critically, part (c) is labeled AO3 with Show that, which should appear only in final sections; Show that is a mid-tier command term (AO2/AO3), and a 3-mark Show that part mid-sequence violates the progression. Part (d) correctly places AO3 at the end, but the structure before it is inverted.

FIX: Restructure as follows: (a) AO1 Define [2 marks] — correct. (b)(i) AO2 Calculate [3 marks] — change to lower-tier command (keep Calculate). (b)(ii) AO2/AO3 Determine [2 marks] — correct. (c) AO2/AO3 Derive/Show that [3 marks] — relabel as Derive or accept as transitional. (d) AO3 Suggest/Explain [2 marks] — correct final tier. Alternatively, merge (b)(i) and (b)(ii) into a single Calculate part [5 marks], then move (c) Show that [3 marks] to middle, and (d) Suggest [2 marks] as final. The key issue: Show that (part c, 3 marks) appears before Suggest (part d, 2 marks), which is acceptable *if* (c) is explicitly AO2/AO3 transitional; relabel (c) from 'AO3' to 'AO2/AO3' to restore ladder integrity.
cross_theme pass

Part (c) genuinely integrates both A.2 and C.2: the momentum/collision context (pellet's 45 m·s⁻¹ approach speed from A.2) is the physical input, and the Doppler effect (C.2) is the computational core—the pellet's velocity directly determines both frequency shifts, making neither topic dispensable. Part (d) reinforces this by asking candidates to explain mechanical (force, rebound angle) and acoustic (air drag on speed, Doppler geometry) effects separately.
discriminator pass

Part (d) explicitly requires the student to critique the model by suggesting physical reasons why calculated and measured values diverge. This is a genuine model-critique task: students must identify assumptions (purely horizontal rebound, negligible collision time, ideal pellet speed, perfect transducer alignment) and explain how real-world violations cause discrepancies. It is not a calculation.
rule_of_one pass

Each part's mark count aligns with its award criteria. Part (a): 2 marks, 2 bullets; (b)(i): 3 marks, 3 bullets (Δp calculation, rate formula, final answer with units); (b)(ii): 2 marks, 2 bullets; (c): 3 marks, 3 bullets (first Doppler shift, second shift, numerical result); (d): 2 marks, 2 bullets (one reason per discrepancy). All match within tolerance.
topic pass

Part (c) genuinely requires both topics: it applies the Doppler effect (C.2 Wave model) to a scenario where the pellet's velocity is derived from momentum/collision analysis (A.2 Forces and momentum). The two-step Doppler calculation (moving observer, then moving source) is intrinsically tied to the pellet's known approach velocity from the collision setup. Part (d) further integrates both by asking for physical explanations of discrepancies in force (momentum-based) and frequency shift (wave-based) measurements.

Iteration 2 — failed

ao_ladder pass

The ERQ correctly obeys the cognitive ladder: Part (a) is AO1 (Define), parts (b)(i)–(b)(ii) are AO2/AO3 (Calculate/Determine), part (c) is AO2/AO3 (Show that, bridging topics), and part (d) is AO3 (Suggest with discriminator logic). The progression from knowledge through application to evaluation is maintained.
cross_theme fail

Part (c) nominally invokes C.2 (wave model: frequency, wavelength, speed of sound, audible range), but the physics is entirely driven by A.2 (impact rate from momentum transfer). The wave concepts are applied only as post-hoc arithmetic (λ = v/f) to a frequency determined by the ball-firing rate; there is no genuine two-way coupling where wave physics constrains or informs the momentum analysis, nor vice versa. The audibility discussion is window-dressing rather than integral use of C.2.

FIX: Restructure part (c) to require students to use wave physics (e.g., interference or resonance) to predict or explain the impact rate, OR replace with a question where the sound wavelength and diffraction/reflection from the wall affect the observable impact force or momentum transfer. Alternatively, redesign to ask how the repeated pulse train creates standing waves in the room and relate the room's resonant frequencies back to constraints on the launching mechanism—this would genuinely blend both themes.
discriminator pass

Part (d) explicitly asks students to 'Suggest two reasons for this discrepancy' between calculated and measured force, requiring critique of the model assumptions (e.g. perfectly elastic collision, ideal horizontal impact, complete momentum transfer). This is a genuine model-critique question, not a calculation.
rule_of_one fail

Part (c) lists 3 marks but the award structure is problematic: M1 awards 1 mark for frequency (12 Hz), but M2 is a complex conditional criterion that conflates audibility reasoning (which should be separate from wavelength calculation), and M3 awards the wavelength. The mark distribution does not cleanly align 3 distinct operations. Additionally, M2's conditional logic ('fundamental is sub-audible BUT harmonics are audible OR broadband transient is audible') creates ambiguity about whether a candidate earns M2 for correctly identifying 12 Hz as sub-audible, or only if they go further to discuss harmonics/transients—this violates clarity of award criteria.

FIX: Restructure Part (c) [3 marks] as: M1 = Calculate f = 12 Hz from 12 impacts/s [1 mark]; M2 = Explain that while the fundamental (12 Hz) is below the audible threshold (~20 Hz), the short impulsive nature of impacts produces audible high-frequency content (harmonics or broadband spectrum) [1 mark]; M3 = Calculate λ = v/f = 340/12 ≈ 28 m for the fundamental (or for a specific stated audible harmonic frequency), with correct unit and value [1 mark]. This cleanly separates frequency calculation, audibility reasoning, and wavelength calculation into 3 distinct marks.
topic pass

Part (c) genuinely integrates both topics: it requires A.2 concepts (impact rate = 12 Hz derived from momentum and collision dynamics) and C.2 concepts (wave model: frequency, wavelength calculation via λ = v/f, and audible frequency range). Neither topic is superficial; both are essential to answer the question completely.

Planner output (stage 1)

Archetype: real-world device
Scenario: A tennis ball machine fires balls horizontally at a wall. Each ball has mass 0.057 kg and leaves the machine at speed 20 m·s⁻¹. The machine fires 12 balls per second continuously. After striking the wall, balls either stick (inelastic collision) or bounce back elastically depending on wall condition. Students measure the force exerted by the stream of balls on the wall and investigate how collision type affects momentum transfer. The scenario bridges classical momentum and elastic/inelastic processes.
Cross-pollination part: d
Discriminator part: d

Part	Marks	AO	Command
a	2	AO1\|AO2	Define
b(i)	3	AO2\|AO3	Calculate
b(ii)	2	AO2	Determine
c	3	AO2\|AO3	Show that
d	2	AO3	Suggest

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