Physics • Year 11 • Module 2: Dynamics • Lesson 6

Work and Kinetic Energy

Build HSC Band 5–6 extended-response technique by analysing multi-variable scenarios, evaluating real-world implications of the work-energy theorem, and designing investigations involving energy.

Master · Extended Response

1. Data + scenario: crumple zones and the work-energy theorem (Band 5–6)

8 marks Band 5–6

Scenario. Modern cars are engineered with “crumple zones” — front and rear sections designed to deform progressively in a collision, increasing the distance over which the car decelerates from its initial speed to zero. The table below compares a 1400 kg car at 60 km/h (16.67 m/s) stopping due to: (A) an ideal rigid barrier (no deformation, instantaneous stop), and two crumple-zone scenarios.

Scenario	Initial speed (m/s)	Deceleration distance (m)	Braking force (N)	Time to stop (s)
A — Rigid barrier	16.67	0.001 (essentially 0)	Calculated	≈ 0
B — Small crumple zone	16.67	0.30	Calculated	≈ 0.036
C — Full crumple zone	16.67	0.60	Calculated	≈ 0.072

Illustrative data. Mass = 1400 kg. Car decelerates from 16.67 m/s to rest in each case. Use W_net = ΔKE to find braking force for each scenario.

Q1. Analyse and evaluate the data above to explain how crumple zones reduce collision forces, and assess the importance of the work-energy theorem as a tool for understanding vehicle safety design. In your response you must:

Calculate the initial kinetic energy of the car and the braking force for each of the three scenarios.
Explain, using W_net = ΔKE, why a longer deceleration distance requires a smaller braking force to achieve the same change in KE.
Compare the braking forces in scenarios A, B, and C and discuss the physiological significance for the driver (a human body can withstand approximately 300 000 N before fatal injury).
Evaluate how the work-energy theorem, rather than Newton’s Second Law alone, provides a more direct route to understanding energy dissipation in crashes.
State one limitation of the crumple-zone model presented here.

Stuck? Plan: KE = ½×1400×16.67² ≈ 194 900 J. Force = KE / deceleration distance (from −F×s = −KE). Scenario A: F = 194 900 / 0.001 = ≈195 000 000 N. B: 194 900 / 0.30 ≈ 649 700 N. C: 194 900 / 0.60 ≈ 324 900 N. Compare each against the 300 000 N survival threshold.

2. Experimental design — verifying the work-energy theorem with a trolley (Band 5–6)

7 marks Band 5–6

Research question. A student claims that “you can’t measure work directly, so the work-energy theorem can’t really be tested experimentally.” Design a laboratory investigation to test whether the work done by a constant net horizontal force on a trolley equals the change in kinetic energy of that trolley.

Constraints: You have access to a trolley (0.5–2.0 kg), a ramp/track, a hanging mass on a string over a pulley to provide constant force, a metre ruler, a stopwatch (or a light gate / motion sensor), and standard masses. Your investigation should take no longer than one laboratory session (60 minutes).

Q2. Design the investigation and present it in the format below.

State your hypothesis (a testable prediction linking W_net and ΔKE, including the independent and dependent variables).
Identify the independent variable, dependent variable, and at least two controlled variables.
Describe the procedure in at least five numbered steps, including how you measure both work done and change in KE independently.
State what result would falsify your hypothesis.
Identify two sources of systematic error and explain how each would affect the result.

Stuck? Measure work directly as W = F×s (tension in string × displacement of trolley; measure F with a Newton meter and s with a ruler). Measure KE independently as KE = ½mv² (measure final speed v with a light gate, or use v = 2s/t for constant acceleration from rest). If W_net ≠ ΔKE, the theorem is falsified. Systematic errors: friction on the track reduces W_net; the hanging mass also accelerates, meaning T ≠ mg (overestimate of force).

Answers — Do not peek before attempting

Q1 — Sample Band 6 response (8 marks), annotated

KE and braking forces: KE = ½ × 1400 × 16.67² = 700 × 277.9 = 194 530 J ≈ 195 000 J [1 — correct KE calculation]. Scenario A: F_b = KE/s = 195 000/0.001 = 195 000 000 N (195 MN) [1]. Scenario B: F_b = 195 000/0.30 = 650 000 N (650 kN) [1]. Scenario C: F_b = 195 000/0.60 = 325 000 N (325 kN) [1].

W_net = ΔKE explanation: The work-energy theorem states that W_net = ΔKE = F_b×s (in magnitude). For the same initial KE (fixed ΔKE), the product F_b×s must remain constant. If s increases, F_b must decrease proportionally. Doubling s halves F_b; this is the engineering basis of crumple zones [1].

Physiological significance: Scenario A produces a force of ~195 MN — orders of magnitude beyond any survivable threshold (~300 000 N). Scenario B (~650 kN) still exceeds the threshold. Scenario C (~325 kN) just barely exceeds the threshold; real crumple zones typically aim for 0.3–0.9 m deformation to reduce peak force below lethal levels. The comparison shows that even a small crumple zone dramatically reduces force, but adequate deformation length is critical [1].

Why W-E theorem over Newton’s 2nd law: Newton’s Second Law (F = ma) requires knowing the acceleration throughout the collision, which varies rapidly and is difficult to measure in a real crash. The work-energy theorem requires only the initial and final speeds and the deceleration distance — all measurable from crash test data. It directly links the energy that must be dissipated to the design distance, making it the preferred engineering tool for crash analysis [1].

Limitation: The model assumes constant braking force and treats the car as a rigid point mass, ignoring the complex deformation pattern of real crumple zones, the variable stiffness of crumple zones as they compress, and the energy absorbed by the human body (not just the car structure). In reality, peak force varies and the “effective” deceleration distance may be less than the total crumple length [1].

Marking criteria (8 marks): 1 = correct KE calculation (accept ≈195 000 J); 1 = correct F for Scenario A; 1 = correct F for Scenario B; 1 = correct F for Scenario C; 1 = clear explanation of the inverse relationship between F and s at constant ΔKE (W-E theorem); 1 = physiological comparison using the 300 000 N threshold with correct interpretation of which scenarios are survivable; 1 = explicit explanation of why the W-E theorem is more practical than F=ma for crash analysis; 1 = one valid limitation of the model.

Q2 — Sample Band 6 response (7 marks), annotated

Hypothesis: If the work-energy theorem holds, the net work done on the trolley (W_net = F_net×s, measured as tension force times displacement) will equal the change in kinetic energy (ΔKE = ½mv_f² − 0) to within experimental uncertainty. Independent variable: applied force (varied by changing the hanging mass). Dependent variable: final kinetic energy of the trolley (measured via final speed). Controlled variables: trolley mass (same trolley throughout); track surface; total displacement of trolley. [1 — testable hypothesis with IV and DV]

Procedure: (1) Measure the mass of the trolley (m) on a digital balance. Connect a string over a pulley to a hanging mass (M) to provide a constant tension T ≈ Mg. Measure T directly with a Newton meter. (2) Set the trolley at rest and mark a starting position. Mark a finish line at distance s = 1.000 m from the starting line. (3) Release the trolley from rest; use a light gate at the finish line to record the final speed v_f. (Alternatively, time the trolley over the 1 m with a stopwatch and use v_f = 2s/t for constant acceleration from rest.) (4) Calculate W_net = T × s and ΔKE = ½mv_f². Record in a table. (5) Repeat with three different hanging masses (e.g. 50 g, 100 g, 200 g), recording T, v_f, W_net, and ΔKE for each trial. Repeat each mass condition three times and average. [1 — five steps including independent KE measurement]

Falsification: If W_net consistently differs from ΔKE by more than experimental uncertainty (e.g. by >10%), the hypothesis would be falsified — indicating that energy is being lost to an unaccounted source (e.g. friction) or that the force measurement is incorrect. A systematic bias of W_net > ΔKE in every trial would indicate energy is being lost (likely to friction), which could itself be investigated as a follow-up. [1]

Systematic errors: (1) Friction on the track: the track is not perfectly frictionless, so the net force on the trolley is less than T (tension minus friction). This means T overestimates F_net, so W_net as calculated will be larger than ΔKE — the theorem will appear to fail on the side of excess work. Improvement: measure friction force separately (pull the trolley at constant speed with a Newton meter) and subtract it from T. (2) The hanging mass also accelerates: the standard assumption T = Mg holds only when M << m (trolley mass). If M is comparable to m, T < Mg because the hanging mass accelerates. This causes an overestimate of T and therefore of W_net. Improvement: use hanging masses much smaller than the trolley mass (<10%), or use the full Atwood’s machine equation T = Mm g/(M + m). [1 per systematic error = 2]

Marking criteria (7 marks): 1 = testable hypothesis with explicit IV and DV linked to W-E theorem; 1 = controlled variables (at least two named and justified); 1 = five clear procedural steps including measurement of both W and ΔKE independently; 1 = valid falsification statement (numerical threshold or direction of discrepancy); 1 = systematic error 1 identified with effect on result; 1 = systematic error 2 identified with effect on result; 1 = precise physics terminology throughout (net work, kinetic energy, tension, displacement, v²).