Generated ERQ

✓ passed A.2 Forces and momentum × E.3 Radioactive decay 12 marks HL 2 passes 78.05s $0.4580

## ERQ · 12 marks · Topics: A.2 Forces and momentum + E.3 Radioactive decay **Stem.** A research team studies the recoil of phosphorus-32 daughter nuclei following beta-minus decay: ³²P → ³²S + e⁻ + ν̄ₑ. The parent ³²P nucleus is at rest inside an ultracold ion trap. The decay has an endpoint kinetic energy (Q-value) of 1.71 MeV, shared between the electron and the antineutrino; the recoiling ³²S nucleus takes a negligible share of the energy but carries away momentum. In one detected event, the emitted electron is measured to have kinetic energy 0.50 MeV and travels in the +x direction; the antineutrino is emitted in the −x direction. The recoil ³²S ion (mass 5.31 × 10⁻²⁶ kg) is collected on a microchannel plate located 12 cm from the trap. Take mₑ = 9.11 × 10⁻³¹ kg, c = 3.00 × 10⁸ m s⁻¹, 1 eV = 1.60 × 10⁻¹⁹ J. Treat the antineutrino as massless so that its momentum is p_ν = E_ν / c. ### Part (a) State [2 marks] · AO1 · Topic: E.3 + A.2 State the law of conservation of linear momentum, and state the total vector momentum of the three decay products (electron, antineutrino, recoil nucleus) immediately after the decay of the ³²P nucleus. ### Part (b)(i) Calculate [3 marks] · AO2/AO3 · Topic: E.3 For this event, calculate the magnitude of the antineutrino's momentum, in kg m s⁻¹. (Use energy conservation between the Q-value and the kinetic energies of the electron and antineutrino; the nuclear recoil energy is negligible.) ### Part (b)(ii) Determine [3 marks] · AO2/AO3 · Topic: A.2 + E.3 The electron in this event is relativistic. Using the relativistic relation E² = (pc)² + (mc²)², determine the magnitude of the electron's momentum, in kg m s⁻¹. ### Part (c) Determine [2 marks] · AO3 · Topic: A.2 + E.3 Using conservation of momentum applied to the three-body decay (with the electron in +x and the antineutrino in −x), determine the recoil speed of the ³²S nucleus and state its direction. ### Part (d) Suggest [2 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR In a repeated run of identical decays, the team observes that the measured recoil speeds of the ³²S ions form a broad distribution rather than the single value calculated in (c). Suggest **two** distinct reasons, rooted in the physics of beta decay, why a single predicted value fails to describe the experimental distribution. --- ## Mark Scheme ### Part (a) [2 marks] - M1: Total momentum of an isolated system remains constant / the vector sum of momenta before = vector sum after (in absence of external forces) [ECF: no] - M2: Since parent ³²P is at rest, total momentum of (e⁻ + ν̄ₑ + ³²S) = 0 (vector sum is zero) [ECF: no] ### Part (b)(i) [3 marks] - M1: Energy conservation: E_ν = Q − KE_e = 1.71 − 0.50 = 1.21 MeV [ECF: no] - M2: Convert and apply p_ν = E_ν/c: p_ν = (1.21 × 10⁶ × 1.60 × 10⁻¹⁹) / (3.00 × 10⁸) [ECF: yes] - M3: p_ν ≈ 6.45 × 10⁻²² kg m s⁻¹ (accept 6.4–6.5 × 10⁻²², 2–3 sf, correct units) [ECF: yes] ### Part (b)(ii) [3 marks] - M1: Identify total energy E = KE + mₑc² = 0.50 + 0.511 = 1.011 MeV (accept 1.01 MeV) [ECF: no] - M2: pc = √(E² − (mₑc²)²) = √(1.011² − 0.511²) ≈ 0.872 MeV, then p_e = (0.872 × 10⁶ × 1.60 × 10⁻¹⁹)/(3.00 × 10⁸) [ECF: yes] - M3: p_e ≈ 4.65 × 10⁻²² kg m s⁻¹ (accept 4.6–4.7 × 10⁻²², 2–3 sf, correct units) [ECF: yes] ### Part (c) [2 marks] - M1: From ΣP = 0 in x: p_S = p_ν − p_e = 6.45 × 10⁻²² − 4.65 × 10⁻²² = 1.80 × 10⁻²² kg m s⁻¹ (sign indicates +x direction since antineutrino momentum in −x dominates) [ECF: yes from (b)(i),(b)(ii)] - M2: v_S = p_S / m_S = 1.80 × 10⁻²² / 5.31 × 10⁻²⁶ ≈ 3.4 × 10³ m s⁻¹ in the +x direction (same direction as the electron) [ECF: yes] ### Part (d) [2 marks] — award any TWO of: - The antineutrino carries a **continuous, variable share** of the Q-value, so the electron and antineutrino momenta differ event-to-event → recoil momentum magnitude varies. [ECF: no] - The decay products are emitted in **3-D with variable relative angles**, not collinear as assumed in (c); the vector sum of p_e and p_ν then has a different magnitude in each event. [ECF: no] - Accept: thermal motion of the parent ion in the trap adds a Maxwell–Boltzmann spread to the recoil; finite detector/timing resolution; relativistic angular correlation between e⁻ and ν̄ₑ (Fermi theory). [ECF: no] ### Marker notes - Alternative for (b)(ii): non-relativistic p = √(2mₑKE) gives ≈ 3.8 × 10⁻²² kg m s⁻¹ — **do not award M2/M3**; KE = 0.50 MeV ≈ mₑc² so relativistic treatment is required (M1 only if E_total identified). - (c): accept negative sign convention provided direction is stated explicitly; full ECF from (b)(i) and (b)(ii). If candidate forgets vector subtraction and adds magnitudes (giving ~2.1 × 10⁴ m s⁻¹), award M1 only if correctly executed arithmetic but lose M2 for missing direction logic. - (d) discriminator: the key distinction is between (i) the **energy-sharing continuum** (E.3 three-body kinematics) and (ii) the **angular distribution** in 3-D — both must be of decay-physics origin; do not credit "measurement error" or "air resistance" as one of the two.

Critic verdicts

Iteration 0 — failed

ao_ladder pass

The ERQ correctly follows the cognitive ladder: (a) is AO1 with State/Outline command terms; (b)(i)–(b)(ii) and (c) are AO2/AO3 with Calculate/Determine/Show that; (d) is AO3 with Suggest. Progression from knowledge→application→analysis→synthesis is proper and internally coherent across both topics.
cross_theme fail

Topic E.3 (radioactive decay) is mentioned only as a name-checked label in part (c) and discussed descriptively in part (d); it is never actually *used* in solving the momentum calculation in (c). Part (c) uses only A.2 conservation of momentum with the assumption that the antineutrino is absent — E.3 content (the three-body nature of beta decay) appears only in the final part (d) as post-hoc explanation, not as an integrated physics concept driving the solution.

FIX: Restructure part (c) to require students to *apply* E.3 concepts during the calculation: for example, ask them to calculate the recoil speed when a *specified fraction* of the endpoint energy is carried by the antineutrino (varying case), or to derive the condition on antineutrino momentum needed to produce a given recoil speed. This would genuinely blend both topics by requiring E.3 physics (three-body decay kinematics) to solve an A.2 momentum problem, rather than treating E.3 as a contextual footnote in part (d).
discriminator pass

Part (d) explicitly asks students to suggest why measured recoil speeds are smaller and distributed continuously, requiring explanation of the physical model (three-body decay, variable antineutrino momentum sharing). This is a genuine model-critique question that discriminates between superficial and deep understanding of beta decay.
rule_of_one pass

All mark counts align correctly with award-criterion bullets: (a) 2 marks / 2 bullets; (b)(i) 3 marks / 3 bullets; (b)(ii) 3 marks / 3 bullets; (c) 2 marks / 2 bullets; (d) 2 marks / 2 bullets. Each Calculate/Determine/Show part has the required formula, substitution, and answer structure.
topic fail

Topic E.3 (Radioactive decay) is mentioned only nominally in the stem and part (d). Parts (a)–(c) test exclusively A.2 (Forces and momentum); they do not require any conceptual understanding of the decay process itself—only that a beta particle exists with given kinetic energy. Part (d) asks students to explain why recoil speeds vary, which touches on the three-body nature of beta decay, but this is a superficial invocation occurring only in the final 2-mark question. The core physics being tested—momentum conservation, impulse, kinematics in the absorber, recoil calculation—all derive from A.2 alone.

FIX: Restructure the question so that understanding the decay mechanism (e.g., Q-value, endpoint energy, mass-energy balance, or neutrino energy) is central to solving at least one of parts (a)–(c). For example: (i) ask students to derive the endpoint energy from the mass difference between parent and daughter nuclei and the antineutrino; (ii) require calculation of the antineutrino momentum in a specific decay event, then use momentum conservation to find both beta and nuclear recoil; or (iii) ask how the continuous energy spectrum affects the momentum distribution and thus the recoil speed distribution. As written, E.3 is a decoration, not a tested topic.

Iteration 1 — passed

ao_ladder pass

The ERQ obeys the cognitive ladder correctly. Part (a) is AO1 with 'State' command term (conservation law + vector momentum). Parts (b)(i)–(b)(ii) and (c) are AO2/AO3 with 'Calculate/Determine' commands involving energy conservation, relativistic kinematics, and momentum conservation. Part (d) is AO3 with 'Suggest' command term requiring explanation of physical reasons for distribution. Difficulty and assessment objectives progress logically from recall through procedural calculation to higher-order reasoning about experimental scatter.
cross_theme pass

The ERQ genuinely blends A.2 (Forces and momentum) and E.3 (Radioactive decay) throughout. Part (b)(i) uses energy conservation specific to beta decay; part (b)(ii) applies relativistic momentum via the decay kinematics; part (c) uses conservation of momentum to find recoil speed; part (d) requires understanding of three-body decay physics. Both concepts are essential to solving the problem, not merely name-checked.
discriminator pass

Part (d) explicitly asks students to 'Suggest two distinct reasons, rooted in the physics of beta decay, why a single predicted value fails to describe the experimental distribution.' This requires critique of the model assumptions (collinearity, fixed energy-sharing) and explanation of physical limitations, not further calculation.
rule_of_one pass

All parts have mark counts that match their award-bullet counts. Parts (a), (b)(i), (b)(ii), and (c) each have 2 or 3 marks with exactly 2 or 3 distinct operations listed. Part (d) awards 2 marks from a pool of 4 options (correctly stated as 'any TWO'), which is acceptable for a 'suggest' question where multiple valid physics reasons exist. The structure complies with the dimension's contract.
topic pass

The ERQ rigorously tests both A.2 (Forces and momentum) and E.3 (Radioactive decay) across all parts. Part (a) invokes conservation of momentum (A.2) explicitly applied to a three-body beta decay (E.3). Parts (b)–(c) require energy conservation from decay physics (E.3) combined with relativistic momentum relations and vector momentum balance (A.2). Part (d) demands explanation of the continuous recoil spectrum rooted in three-body decay kinematics (E.3) and variable momentum vectors (A.2). Neither topic is superficial; both are essential to solving the problem.

Planner output (stage 1)

Archetype: real-world device
Scenario: A radioactive sample undergoes beta decay inside a sealed cylindrical container. The emitted beta particles (electrons) experience a retarding force from air molecules and a magnetic field perpendicular to their initial direction of motion. Students must analyse the momentum transfer during particle deceleration, relate it to the impulse delivered by the magnetic and frictional forces, and then evaluate how the continuous nature of beta decay and the energy spectrum of emitted particles affect the validity of treating momentum change as a simple two-body collision.
Cross-pollination part: d
Discriminator part: d

Part	Marks	AO	Command
a	2	AO1	State the definition of momentum and outline what happens to the momentum of a beta particle as it slows down in air.
b(i)	3	AO2	Calculate the momentum change of a beta particle that is emitted with initial kinetic energy 2.0 MeV and comes to rest after traveling 5 mm through air (mass of electron = 9.11 × 10⁻³¹ kg; 1 eV = 1.6 × 10⁻¹⁹ J).
b(ii)	3	AO2	Determine the average retarding force acting on the beta particle during its deceleration, assuming the deceleration is uniform.
c	2	AO3	Show that the impulse delivered to the air molecules equals the magnitude of momentum change of the beta particle, using Newton's third law.
d	2	AO3	Evaluate how the continuous energy spectrum of beta decay (emission of antineutrinos carrying variable energy) and the random nature of decay events affect the assumption that individual momentum transfers can be treated as isolated two-body collisions.

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