HSC Exam Practice

Sample response. Use conservation of mechanical energy (KE₁ + U₁ = KE₂ + U₂) only when the system is frictionless and only gravity does work. Use the work-energy theorem (W_net = ΔKE) when friction or any other non-conservative force is present. The physical reason is that friction converts mechanical energy to heat, so mechanical energy is not conserved — the W_net = ΔKE formula accounts for all work done including work done against friction, which is negative.

Marking notes. 1 = states the condition (frictionless/only gravity → use conservation); 1 = states the alternative (friction present → use W_net = ΔKE); 1 = explains why friction rules out conservation (mechanical energy is converted to heat, not preserved).

Sample response. (1) W = Fs cosθ — use when calculating energy transferred by a single force through a displacement (e.g. finding work done by an engine over a distance). (2) KE = ½mv² — use when finding the energy of motion at any instant (e.g. finding KE of a car at the bottom of a hill). (3) W_net = ΔKE — use when friction is present and you need to find the change in speed (e.g. block sliding down a rough ramp). (4) ΔU = mgΔh — use when an object changes vertical height in a gravitational field (e.g. calculating PE at the top of a hill). (5) KE₁ + U₁ = KE₂ + U₂ — use when the system is frictionless and only gravity acts (e.g. free fall or frictionless slope). (6) P = ΔE/Δt = Fv — use when finding the rate of energy transfer or connecting force, speed, and power (e.g. engine power needed to maintain constant speed).

Marking notes. 1 mark per formula named with a correct applicable situation. Accept any valid situation consistent with the formula’s conditions.

Sample response. W = Fs cosθ. The normal force acts vertically upward while the object’s displacement is horizontal. The angle between the normal force and the direction of motion is therefore 90°. Since cos 90° = 0, the work done is W = Fs × 0 = 0 regardless of the magnitude of the normal force or the distance moved. The normal force is perpendicular to displacement and does no work.

Marking notes. 1 = quotes W = Fs cosθ and states θ = 90° between normal force and displacement; 1 = evaluates cos90° = 0 → W = 0; 1 = explains the geometry (normal is vertical, motion is horizontal, so perpendicular).

Sample response. “Energy lost” means that mechanical energy (KE + PE) has decreased, but it has NOT been destroyed — it has been converted to thermal energy (heat) in the surfaces in contact with friction, and sometimes to sound. Energy is never destroyed; the total energy of the system plus surroundings remains constant. This is governed by the law of conservation of energy (first law of thermodynamics). “Energy destroyed” would violate this fundamental law and does not occur.

Marking notes. 1 = distinguishes the terms (lost = converted, not destroyed); 1 = states where the energy goes (thermal/heat energy in surfaces; accept sound); 1 = names the law of conservation of energy (or first law of thermodynamics).

Sample response. The student’s claim is incorrect. From conservation of mechanical energy (frictionless, starting from rest): mgh = ½mv². Mass m appears on both sides and cancels: v = √(2gh). The speed at the bottom depends only on the vertical height h and g — not on the mass. All objects, regardless of mass, reach the same speed at the bottom of a frictionless ramp from the same height. This is analogous to free fall, where all objects accelerate at g regardless of mass.

Marking notes. 1 = correctly identifies the error (mass does not affect the speed); 1 = shows the algebra where mass cancels from mgh = ½mv² to give v = √(2gh); 1 = states the correct relationship: speed depends on height h and g only, not on mass.

Sample response (a). Ideal KE = mgh = 60 × 9.8 × 4.0 = 2352 J (same for all runs, frictionless assumption).

Run 1: KE = ½ × 60 × 8.1² = 1968 J; efficiency = 1968/2352 × 100 = 83.7%.

Run 2: KE = ½ × 60 × 7.9² = 1873 J; efficiency = 79.6%.

Run 3: KE = ½ × 60 × 8.3² = 2067 J; efficiency = 87.9%.

Run 4: KE = ½ × 60 × 8.0² = 1920 J; efficiency = 81.6%.

Run 5: KE = ½ × 60 × 8.2² = 2018 J; efficiency = 85.8%.

Marking notes. 1 = correct ideal KE = 2352 J with working shown; 1 = correct KE calculation for Run 1 (accept 1967–1969 J); 1 = correct efficiency for Run 1 (accept 83–84%); 1 = remaining four rows calculated correctly (1 mark if at least 3 of the remaining 4 rows are correct).

Sample response (b). The efficiency varies between approximately 80% and 88% across runs with no consistent upward or downward trend — the variation is random. One physical reason: slight differences in the skateboarder’s body position on each run change the effective aerodynamic drag. The efficiency is always less than 100% even on an “apparently frictionless” ramp because air resistance (drag) is always present and converts some KE to thermal energy in the air. Additionally, the wheels/bearings have low but nonzero friction.

Marking notes. 1 = correctly describes trend (no consistent pattern / approximately random variation); 1 = names a valid physical reason for run-to-run variation (body position, air resistance magnitude, push-off variation, surface imperfections); 1 = explains why efficiency is always <100% (air resistance or wheel friction always present, even on “frictionless” ramp).

Sample response (c). Average efficiency = (83.7 + 79.6 + 87.9 + 81.6 + 85.8)/5 = 83.7% (accept 83–84%). Total initial energy on run 6: KE_i + PE = ½ × 60 × 2.0² + 2352 = 120 + 2352 = 2472 J. Actual final energy = 0.837 × 2472 = 2069 J. Speed at bottom: v = √(2 × 2069/60) = √(69.0) = 8.31 m/s.

Marking notes. 1 = correct average efficiency stated; 1 = total initial energy = KE_push + mgh = 120 + 2352 = 2472 J; 1 = correct speed at bottom (accept 8.1–8.4 m/s depending on rounding of average efficiency).

Sample Band 6 response. The claim that the six formulae are independent tools is incorrect. Each formula is a different view of the same underlying energy accounting system, and they are connected through the concept of work. W = Fs cosθ is the definition of work — the energy transferred by a force. Dividing by time gives P = Fv (power as rate of work). Setting W_net = ΔKE (the work-energy theorem) connects forces to changes in speed; this is derived directly from Newton’s second law applied over a displacement. When only gravity does work, W_net = −ΔU, which gives the conservation equation KE₁ + U₁ = KE₂ + U₂. And ΔU = mgΔh is simply W = Fs cosθ applied to vertical lifting (with θ = 180° between gravity and upward displacement). The formulae are not parallel — they are nested: each is derived from a simpler one, and each applies under specific conditions that are defined by which forces are present.

Using the wrong formula leads to real errors. For example, applying conservation of mechanical energy to a car descending a rough hill incorrectly ignores the negative work done by friction, predicting a higher speed at the bottom than actually occurs. The error is not merely a calculation mistake — it is a conceptual error about which energy accounting rule applies. Conversely, using W_net = ΔKE on a frictionless hill with initial speed and height produces the correct answer but is more complex than needed — conservation is simpler and exactly equivalent in the frictionless case.

In a real-world synthesis (using the car-on-a-hill scenario from this lesson): a 1200 kg car rolls from rest down a frictionless 20 m hill. Conservation gives v = √(2 × 9.8 × 20) = 19.8 m/s at the bottom — this answer feeds the next step. On the flat road with 900 N resistance and engine power 45 kW, constant velocity requires F_drive = F_resist (Newton 1), giving P = 900 × 19.8 = 17.8 kW needed to maintain hill speed. Reserve power = 45 − 17.8 = 27.2 kW drives acceleration. The ΔKE formula gives the energy needed to accelerate to 30 m/s; dividing by reserve power gives time. The answer from step 1 (v = 19.8 m/s) is the input to step 2. The answer from step 2 feeds step 3. No formula operates independently.

In conclusion, the six formulae are one connected energy accounting system. W = Fs cosθ is the foundation; W_net = ΔKE is the universal rule; conservation and ΔU = mgΔh are special cases of it; P is the rate version; KE = ½mv² is the energy of the state. Understanding the connections, not memorising six separate rules, allows any dynamics problem to be solved.

Marking criteria (9 marks): 1 = explicitly rejects the claim with a clear reason (formulae are connected). 1 = identifies at least two connections between formulae (e.g. W = Fs cosθ → P = Fv; W_net = ΔKE is derived from Newton 2). 1 = explains why using the wrong formula leads to error in a specific example (conservation on rough slope, or vice versa). 1 = names a real-world scenario and correctly applies the synthesis chain (at least three linked steps). 1 = shows how an output feeds the next step in the chain (explicit chain reasoning). 1 = discusses the condition governing conservation vs W_net = ΔKE (frictionless vs friction present). 1 = states role of P = Fv and its connection to Newton 1 at constant velocity. 1 = uses precise physics vocabulary throughout (net work, conservation of mechanical energy, non-conservative force, energy accounting). 1 = reaches an explicit evaluative conclusion that the formulae are one connected system, not six independent tools.