Generated ERQ

✓ passed A.2 Forces and momentum × D.1 Gravitational fields 14 marks HL 3 passes 117.72s $0.7970

``` ## ERQ · 14 marks · Topics: A.2 Forces and momentum + D.1 Gravitational fields · Archetype: theory_application **Integration:** primary=A.2 Forces and momentum, secondary=D.1 Gravitational fields (strength: supporting) **Stem.** A spacecraft of mass 2500 kg is in a circular orbit around Earth at an altitude of 400 km. Mission control plans a Hohmann transfer to raise the spacecraft to a circular orbit at 800 km altitude. The manoeuvre uses two brief thruster burns: the first burn is applied tangentially at the 400 km orbit ("perigee burn") to place the spacecraft on an elliptical transfer orbit whose apogee lies at 800 km altitude; the second burn ("apogee burn") is applied tangentially at apogee to circularise the orbit. Treat both burns as instantaneous (impulsive) so that the spacecraft's position does not change during the burn, only its velocity. Take Earth's mass as 5.97 × 10²⁴ kg, Earth's radius as 6.37 × 10⁶ m, and G = 6.67 × 10⁻¹¹ N m² kg⁻². ### Part (a) State [2 marks] · AO1 · Topic: A.2 State Newton's second law in terms of momentum, and explain why this form is more appropriate than F = ma when analysing a thruster burn. ### Part (b)(i) Calculate [3 marks] · AO2 · Topic: A.2 + D.1 Show that the orbital speed of the spacecraft in the initial 400 km circular orbit is 7.67 × 10³ m s⁻¹. ### Part (b)(ii) Determine [3 marks] · AO2 · Topic: A.2 The perigee burn changes the spacecraft's speed from 7.67 × 10³ m s⁻¹ to 7.84 × 10³ m s⁻¹ (the speed required at perigee of the elliptical transfer orbit). The thruster expels exhaust gas at a speed of 3.10 × 10³ m s⁻¹ relative to the spacecraft. Determine the mass of propellant ejected during the perigee burn. ### Part (c) Derive [4 marks] · AO3 · Topic: A.2 + D.1 By equating the gravitational force to the centripetal force, derive an expression for the total mechanical energy E of a satellite of mass m in a circular orbit of radius r around a planet of mass M. Hence calculate the magnitude of the energy that must be supplied by the apogee burn to circularise the orbit at 800 km altitude, given that the speed of the spacecraft at apogee of the transfer orbit (before the burn) is 7.13 × 10³ m s⁻¹. ### Part (d) Explain [2 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR Explain one way in which the assumption of an instantaneous (impulsive) burn leads to a systematic discrepancy between the predicted final orbit and the orbit actually achieved. --- ## Mark Scheme ### Part (a) [2 marks] - M1: Net force equals the rate of change of momentum, F = Δp/Δt (or dp/dt) [no ECF] - M2: During a thruster burn the spacecraft's mass changes as propellant is expelled, so F = ma (which assumes constant mass) is not strictly valid; the momentum form correctly accounts for the changing mass [no ECF] ### Part (b)(i) [3 marks] — Show that - M1: Equates gravitational force to centripetal force: GMm/r² = mv²/r, giving v = √(GM/r) [no ECF] - M2: Substitutes r = 6.37 × 10⁶ + 0.400 × 10⁶ = 6.77 × 10⁶ m with M = 5.97 × 10²⁴ kg [no ECF] - M3: v = √[(6.67 × 10⁻¹¹ × 5.97 × 10²⁴)/(6.77 × 10⁶)] = 7.67 × 10³ m s⁻¹ (accept 7.66–7.68 × 10³) [no ECF] ### Part (b)(ii) [3 marks] — Determine (Calculate) - M1: Identifies conservation of momentum / rocket equation approach. Using impulse–momentum equivalence: m_prop × v_exhaust = m_spacecraft × Δv (impulsive approximation), OR uses Tsiolkovsky: Δv = v_e ln(m_i/m_f) [no ECF] - M2: Δv = 7.84 × 10³ − 7.67 × 10³ = 170 m s⁻¹; substitution: m_prop = (2500 × 170)/(3.10 × 10³) [ECF from (b)(i)] - M3: m_prop = 1.4 × 10² kg (accept 137–140 kg). Tsiolkovsky route gives m_prop = 2500(1 − e^(−170/3100)) ≈ 133 kg — also accept. [ECF from (b)(i)] ### Part (c) [4 marks] — Derive - M1: From GMm/r² = mv²/r obtains KE = ½mv² = GMm/(2r) [no ECF] - M2: States gravitational PE = −GMm/r and hence E = KE + PE = −GMm/(2r) [no ECF] - M3: At r₂ = 7.17 × 10⁶ m, circular orbital speed v₂ = √(GM/r₂) = 7.45 × 10³ m s⁻¹; computes ΔKE = ½ × 2500 × (7450² − 7130²) [ECF from M1–M2] - M4: ΔE = ½ × 2500 × (7450² − 7130²) ≈ 5.8 × 10⁹ J (accept 5.7–5.9 × 10⁹ J). Equivalent method using E = −GMm/(2r) at both radii also accepted. [ECF from M1–M3] ### Part (d) [2 marks] — Explain (causal chain per §4.4.1) - M1: A real thruster burn has finite duration, so during the burn the spacecraft moves along its orbit and the thrust vector is no longer applied at a single point with purely tangential orientation — part of the thrust acts against gravity ("gravity losses") rather than producing useful tangential Δv [premise / mechanism] - M2: Therefore the impulsive-burn model overestimates the Δv delivered for a given propellant mass, so the actual apogee (and final orbital radius) is systematically **lower** than predicted [explicit "therefore" linking mechanism to orbital consequence] ### Marker notes - (b)(i): Show-that target 7.67 × 10³ m s⁻¹ given to 3 sig figs; student working must show v = √(GM/r) substitution to earn full marks (do not award M3 if only the target value is quoted). - (b)(ii): Accept either the impulsive momentum-conservation method or the Tsiolkovsky rocket equation. ECF applies to Δv if (b)(i) value differs. - (c): Accept direct calculation via E = −GMm/(2r) at both radii: E₂ − E_apogee where E_apogee = ½mv² − GMm/r₂ at the transfer-orbit apogee, giving the same numerical answer to within rounding. - (d) full credit requires BOTH (i) identification of a physical mechanism (gravity losses, finite burn arc, thrust direction rotating away from tangential, mass change during burn affecting acceleration profile) AND (ii) an explicit directional consequence linking that mechanism to the orbit (lower apogee / smaller final radius / orbit falls short of target). Award 1 mark only if the mechanism is stated without the directional orbital consequence, or vice versa. - Accept for (d) M1 alternatives: finite burn means propellant is ejected over a range of positions where g varies; thrust applied while spacecraft rotates around Earth means impulse direction changes. ```

Critic verdicts

Iteration 0 — failed

ao_ladder pass

The ERQ correctly obeys the cognitive ladder. Part (a) is AO1 with 'State' command term; parts (b)(i)–(b)(ii) are AO2 with 'Calculate' and 'Determine'; part (c) is AO3 with 'Derive' and 'Show that'; part (d) is AO3 with 'Explain'. The progression from recall → application → analysis/synthesis is strict and properly ordered across all 14 marks.
command_structure fail

Part (d) fails the §4.4.1 AO3 'Explain' structure requirement. The mark scheme provides two bullets that describe separate aspects (variation of g, and error introduced) but lacks an explicit 'X therefore Y' causal-chain linking point. M1 identifies the premise (g varies), and M2 states a consequence (error increases), but the marking points do not form a single integrated causal chain with explicit logical connectors showing why the assumption breaks down. The structure should explicitly mark one point for the causal mechanism itself.

FIX: Restructure Part (d) marks to enforce explicit causal chaining: M1 should mark identification that g = GM/r² varies significantly over the large radial range of high-altitude transfers. M2 should award a single integrated marking point: 'Therefore [explicit causal word: 'consequently', 'thus', 'because'] treating the burn as instantaneous (constant g at one point) introduces systematic error, since the spacecraft traverses a significant arc during finite-duration burn through regions of appreciably different gravitational field strength / gravitational losses on non-radial components become significant'. This creates a single explicit X→Y causal chain meeting §4.4.1 'Explain' requirements.
cross_theme pass

Part (c) genuinely integrates both topics: the gravitational field concept (D.1) is essential in deriving v = √(GM/r) by equating gravitational force to centripetal force, and momentum/forces (A.2) underpin the centripetal acceleration framework. Part (d) further reinforces D.1 by explicitly requiring analysis of how gravitational field strength variation along the trajectory affects the validity of the A.2 assumption. Neither topic is name-checked only; both are operationally necessary.
discriminator pass

Part (d) explicitly asks students to explain why the 'instantaneous burn' assumption becomes less valid for much higher orbits, requiring them to critique the model by discussing how gravitational field strength variation along the trajectory introduces error. This is a genuine model-critique question, not a calculation.
rule_of_one fail

Part (c) is marked as [4 marks] — Derive with a paired task (derive expression + show that). The mark scheme lists 4 award criteria (M1–M4), which aligns with the mark count. However, the structure violates IB protocol: a single 'Derive [4 marks]' part should contain one derivation task with ~4 steps, not two distinct tasks (derivation + show-that calculation). The 'show that' component should be a separate Calculate or Determine part worth its own marks. This conflates two different AO3/AO2 operations into one part label.

FIX: Split Part (c) into: (c) Derive [2 marks] — derive v = √(GM/r) from F_grav = F_centripetal (M1, M2); then (c)(ii) Calculate [2 marks] — show that v ≈ 7450 m s⁻¹ at 800 km altitude (M3, M4 as substitution + answer). This preserves the 4-mark total and clarifies the two distinct cognitive operations.
topic pass

The ERQ robustly tests both topics. A.2 Forces and momentum is thoroughly addressed in parts (a)–(b) via Newton's third law, momentum conservation, and the impulse–momentum theorem. D.1 Gravitational fields is substantively tested in part (c) through deriving orbital speed from equating gravitational and centripetal forces, and in part (d) by requiring analysis of how gravitational field strength variation affects burn assumptions. Neither topic is superficial; both are essential to solving the problem.

Iteration 1 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder correctly. Part (a) is AO1 (State); parts (b)(i)–(c)(ii) are AO2 (Calculate/Determine/Show that); part (d) is AO3 (Explain). Command terms align with cognitive demand at each tier, and progression from foundational knowledge through application to evaluation is intact.
command_structure fail

Part (d) mark scheme violates §4.4.1 requirement for Explain command term. M1 states a fact/premise but M2's causal chain is incomplete: it uses 'because' to link to gravity losses, but the warrant connecting 'gravity losses' to the systematic error claim (lower apogee) is not explicitly marked as a separate step. The scheme conflates the mechanism (gravity losses) with the consequence (lower apogee) in a single mark, rather than showing the explicit 'X therefore Y' causal chain required by the dimension specification.

FIX: Restructure Part (d) M2 as two explicit sub-marks: (i) M2a: 'During a finite-duration burn, the varying gravitational field and the rotating thrust direction (no longer purely tangential) mean that part of the impulse is dissipated in fighting gravity losses rather than being converted to useful orbital Δv'; (ii) M2b: 'Therefore, the instantaneous-burn model underestimates gravity losses and systematically over-predicts the Δv achieved, resulting in a final apogee (and orbital radius) lower than predicted.' Alternatively, reword the single M2 to explicitly separate premise from consequence: 'M2: Gravity losses reduce the effective Δv; therefore the instantaneous-burn assumption predicts a higher final apogee than actually achieved, introducing a systematic under-prediction error in the orbit.' Ensure marker notes clarify that both the mechanism (gravity losses) and the directional link to the orbital consequence must be present for full credit.
cross_theme pass

The question genuinely integrates both A.2 (Forces and momentum) and D.1 (Gravitational fields). Part (c)(i) explicitly requires deriving the orbital speed formula by equating gravitational force to centripetal force—both concepts are essential to the derivation, not separable. Part (d) then uses the gravitational field strength variation (g = GM/r²) from D.1 to explain why the momentum-impulse model (A.2) fails under realistic conditions. The integration is structural and causal, not merely name-checking.
discriminator pass

Part (d) explicitly asks students to 'Explain why this assumption introduces a systematic error', requiring critique of the instantaneous-burn model. The mark scheme demands a causal chain linking the assumption (instantaneous burn) to its consequence (gravity losses and orbital shortfall), which is a clear model-critique dimension.
rule_of_one pass

Each part's mark count matches the number of distinct award criteria. All 'Calculate' parts (b)(i), (b)(ii), (c)(ii) have 3 marks with 3 bullets; 'State' (a), 'Derive' (c)(i), and 'Explain' (d) have 2 marks with 2 bullets each. The dimension contract is satisfied.
topic pass

Both topics are genuinely and substantively tested. A.2 (Forces and momentum) is the primary driver: parts (a), (b)(i), (b)(ii), and (d) explicitly require impulse, momentum change, and thrust calculations. D.1 (Gravitational fields) is integrated as the secondary topic: part (c)(i) requires deriving orbital speed by equating gravitational force to centripetal force, and part (c)(ii) applies this to calculate orbital speed at 800 km altitude. Part (d) synthesises both topics by explaining how finite-duration burns introduce gravity losses—a phenomenon rooted in the variation of gravitational field strength during the arc traversed. Neither topic is superficial; each is essential to solving the complete scenario.

Iteration 2 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder correctly. Part (a) is AO1 (State); parts (b)(i)–(b)(ii) are AO2 (Show that / Determine); part (c) is AO3 (Derive with calculation); part (d) is AO3 (Explain with causal mechanism). The progression from recall → application → synthesis is intact, and command terms align with their assigned cognitive levels throughout.
command_structure fail

Part (d) is marked as 'Explain' (AO3) but the mark scheme does NOT explicitly exhibit the required 'X therefore Y' causal-chain structure as a single integrated marking point. Per §4.4.1, an Explain command MUST have at least one explicit causal-chain marking point; the scheme splits this into M1 (mechanism only) and M2 (consequence only), which violates the contract. Additionally, M1 awards a mark for stating a mechanism in isolation ('part of the thrust acts against gravity'), which under §4.2 should not be a standalone marking point without the 'therefore' linkage already present in that mark.

FIX: Restructure part (d) mark scheme as a single causal-chain M1: 'A real thruster burn has finite duration, so the spacecraft moves along its orbit during the burn and the thrust vector rotates away from purely tangential orientation — part of thrust is lost to gravity losses — therefore the actual apogee is systematically lower than predicted.' Award this as one 1-mark point. Then add M2 for a second acceptable mechanism with its own 'therefore' consequence (e.g., mass ejection is spread over a range of orbital positions where gravitational field strength varies, therefore the effective Δv is reduced and final orbit is lower). This ensures each AO3 marking point contains the causal chain explicitly, per §4.4.1.
cross_theme pass

The ERQ successfully integrates both A.2 (Forces and momentum) and D.1 (Gravitational fields). A.2 is primary and runs throughout: Newton's second law (part a), momentum conservation in the rocket equation (b)(ii), and energy change from velocity difference (c). D.1 genuinely supports the solution: gravitational force equating to centripetal force (b)(i) and (c) is essential to finding orbital speeds and deriving the mechanical energy expression. Neither topic is name-checked as filler; both are active in the physics solution pathway.
discriminator pass

Part (d) explicitly asks students to explain how the instantaneous-burn assumption causes a systematic discrepancy between predicted and actual orbits. The mark scheme requires identification of a physical mechanism (e.g., gravity losses, finite burn duration) AND a causal link to the orbital consequence (e.g., actual apogee lower than predicted). This is a genuine model-critique question that probes understanding of where the idealization breaks down.
rule_of_one fail

Part (c) declares 4 marks but lists only 4 award criteria (M1–M4), which appears to pass the count test. However, the part is explicitly bifurcated: "derive an expression" (M1–M2, conceptual, ~2 marks) plus "calculate the magnitude of energy" (M3–M4, computational, ~2 marks). The mark scheme bundles both operations into a single 4-mark part without clearly delineating the derive/calculate boundary. M3 should be isolated as a separate "calculate" sub-part (3 marks: formula, substitution, answer) to match IB convention, leaving the derive as 2 marks. The current structure conflates two distinct cognitive domains.

FIX: Split part (c) into (c)(i) Derive [2 marks] covering M1–M2 only (expression for E in circular orbit); and (c)(ii) Calculate [3 marks] covering M3–M4 plus a final answer-with-units mark for the energy to be supplied. Alternatively, if keeping (c) as a single 4-mark part, retain M1–M4 but explicitly label the first two as 'derive' and the second two as 'calculate' subsections in the part stem.
topic pass

The ERQ successfully integrates both topics throughout: A.2 Forces and momentum is the primary driver (parts a, b, c derive orbital mechanics from F = dp/dt, momentum conservation in rocket equation, and energy methods), while D.1 Gravitational fields provides essential scaffolding (equating gravitational to centripetal force in b(i) and c, computing orbital speeds and gravitational PE). Both topics are genuinely required—neither is merely nominal.

Planner output (stage 1)

Archetype: real-world device
Scenario: A spacecraft of mass 2500 kg is in a circular orbit around Earth at altitude 400 km. Mission control plans a Hohmann transfer to a higher orbit at 800 km altitude. The manoeuvre requires two brief thruster burns: one at perigee (lower orbit) to raise apogee, and one at apogee (higher orbit) to circularise. Students analyse momentum and energy changes during these burns, then evaluate how gravitational field strength variation affects the transfer efficiency.
Cross-pollination part: c
Discriminator part: d

Part	Marks	AO	Command
a	2	AO1	State
b(i)	3	AO2	Calculate
b(ii)	3	AO2	Determine
c	4	AO3	Derive
d	2	AO3	Explain

Generate another →