Physics • Year 11 • Module 2: Dynamics • Lesson 11
Momentum and Impulse
Build HSC Band 5–6 extended-response technique on impulse, direction reversals, safety device evaluation, and experimental design.
1. Data + scenario: helmet liner comparison (Band 5–6)
8 marks Band 5–6
Scenario. A cyclist (head mass = 5.0 kg) falls and their head impacts the ground. Without a helmet the stopping time is 4 ms; with a foam helmet liner the stopping time extends to 80 ms. In both cases the head decelerates from 7.7 m/s to rest. The table below summarises the data.
| Measurement | No helmet | Foam helmet liner |
|---|---|---|
| Head mass (kg) | 5.0 | 5.0 |
| Speed at impact (m/s) | 7.7 | 7.7 |
| Final speed (m/s) | 0 | 0 |
| Stopping time Δt (ms) | 4 | 80 |
| Change in momentum Δp (N s) | calculate | calculate |
| Average force on head (N) | calculate | calculate |
Head impact speed calculated from v = √(2gh) where h = 3.0 m. Illustrative data.
Q1. Analyse and evaluate the experimental data to assess the effectiveness of the helmet liner in reducing injury risk. In your response you must:
- Define positive direction and apply the Vector Protocol. Calculate Δp and average force for both scenarios, showing full working.
- Use the impulse-momentum theorem to explain why the helmet reduces the average force even though Δp is identical in both cases.
- Calculate the ratio of forces (no helmet : with helmet) and explain what this ratio means for brain injury risk.
- Explain one limitation of the simplified model used here (e.g. constant force assumed) and suggest a more accurate approach.
- Evaluate whether a helmet liner that extends stopping time to 40 ms (half the full liner) would provide adequate protection, using quantitative reasoning.
2. Experimental design — testing the impulse-momentum theorem (Band 5–6)
7 marks Band 5–6
Research question. A student claims that the impulse delivered to a trolley on a frictionless track can be measured using a force sensor, and that this should equal the change in momentum of the trolley measured using a motion sensor. Design a controlled investigation to test whether J = Δp holds for a single collision event.
Constraints: Available equipment: dynamics trolley (mass ~0.5 kg), elastic band or spring for impact, calibrated force sensor (datalogged, 1 kHz sampling), motion sensor, frictionless air track or low-friction wheels, digital balance, metre rule, computer with data-logging software.
Q2. Design the investigation and present it in the format below.
- State your hypothesis as a testable prediction linking J (measured from F-t graph area) to Δp (measured from velocity data).
- Identify the independent, dependent, and at least two controlled variables.
- Describe the procedure in at least five numbered steps, including how you will calculate J from the F-t graph and Δp from velocity data.
- State what result would falsify your hypothesis, and explain what a systematic discrepancy (e.g. J consistently 10% greater than Δp) might indicate.
- State two sources of experimental error and one strategy to improve accuracy.
Q1 — Sample Band 6 response (8 marks), annotated
Vector Protocol and calculations [2 marks]: Define downward = positive (direction of head motion). vi = +7.7 m/s; vf = 0. Δp = m(vf − vi) = 5.0(0 − 7.7) = −38.5 N s (same for both scenarios). No helmet: Δt = 0.004 s; F = −38.5/0.004 = −9625 N. Helmet liner: Δt = 0.080 s; F = −38.5/0.080 = −481 N. [1 = Vector Protocol applied + Δp correct; 1 = both forces calculated correctly]
Why the helmet reduces force [1 mark]: The change in momentum is identical in both cases because the head undergoes the same change in velocity (−7.7 m/s to 0). By J = FΔt = Δp, if Δp is fixed and Δt increases by a factor of 20 (4 ms → 80 ms), then F must decrease by a factor of 20. The helmet does not reduce Δp; it extends Δt by compressing, which reduces F.
Force ratio [1 mark]: Ratio = 9625/481 = 20:1. The head experiences 20 times more force without the helmet. Brain injury (concussion, diffuse axonal injury) is closely linked to peak deceleration force; 9625 N applied over 4 ms is almost certainly fatal or causes severe traumatic brain injury, while 481 N over 80 ms is survivable.
Limitation of the model [1 mark]: The model assumes a constant (rectangular) force pulse. In reality, the force rises steeply to a peak and then falls as the foam compresses and then decompresses — more like a triangular or bell-shaped pulse. The peak force can be significantly higher than the average force. A more accurate approach is to integrate the actual F-t curve (area under the curve) from force sensor data, which gives the true impulse without assuming constant force.
40 ms liner evaluation [1 mark]: F = −38.5/0.040 = −963 N. This is 10× less than no helmet (adequate improvement) but only half as effective as the full 80 ms liner (963 N vs 481 N). Whether 963 N is “adequate” depends on injury thresholds: 963 N sustained over 40 ms may still cause concussion for some individuals. The 80 ms liner provides greater safety margin. The 40 ms liner is better than nothing but inferior to the full design. [Full marks require a quantitative comparison of all three forces.]
Marking criteria (8 marks): 1 = defines positive direction; 1 = correct Δp (same for both, −38.5 N s); 1 = both average forces correct; 1 = explains force reduction via J = FΔt = Δp with Δt identified as the changing variable; 1 = correct ratio (20×) with injury risk link; 1 = valid limitation of constant-force assumption with improvement; 1 = correct 40 ms force calculated (963 N); 1 = evaluative judgement on adequacy of 40 ms liner with quantitative reasoning.
Q2 — Sample Band 6 response (7 marks), annotated
Hypothesis [1 mark]: If the impulse-momentum theorem holds, the impulse J measured as the area under the F-t curve from the force sensor will equal the change in momentum Δp = m(vf − vi) measured by the motion sensor, within experimental uncertainty. IV = collision speed (controlled by push). DV = ratio J/Δp. Controlled: trolley mass (same mass each run), surface (same air track), elastic band (same band, same compression).
Procedure [1 mark for five clear steps including calculation methods]: (1) Set up air track horizontal; attach force sensor to fixed end; mount motion sensor at the other end to measure trolley velocity before and after collision. (2) Measure trolley mass with digital balance. Record m. (3) Push trolley toward fixed end at a measured speed; use motion sensor to record vi (just before) and vf (just after) the collision. Calculate Δp = m(vf − vi). (4) Simultaneously record the F-t curve from the force sensor (1 kHz sampling). Integrate numerically (or use software area tool) to find J = ∫F dt over the contact time. (5) Compare J and Δp for each trial. Repeat 5 times; calculate mean ratio J/Δp and standard deviation.
Falsification [1 mark]: If J and Δp are consistently different by more than the measurement uncertainty (i.e. J/Δp differs from 1 by more than ~5%), the hypothesis is falsified. A systematic discrepancy where J > Δp by ~10% would suggest that friction on the track removes impulse (reducing Δp) without appearing in the force sensor reading, or that the force sensor is not aligned with the direction of motion.
Sources of error [2 marks]: (1) Residual friction on the track means some impulse is removed from the trolley by friction (not captured by the force sensor), making Δp less negative than expected and creating a systematic underestimate of Δp relative to J. (2) Misalignment of force sensor: if the sensor is not exactly parallel to the track, only the component of force along the track is measured, underestimating J.
Improvement [1 mark]: Use a well-levelled air track to minimise friction; include a friction correction by measuring the deceleration of the trolley on the track without collision, calculate the friction force, and subtract the friction impulse from Δp before comparison.
Marking criteria (7 marks): 1 = testable hypothesis linking J (area under F-t) to Δp (mass × velocity change); 1 = IV, DV, and two controlled variables named; 1 = five procedural steps including both J and Δp calculation methods; 1 = falsification criterion stated; 1 = explains what a systematic discrepancy means; 1 = two valid error sources; 1 = specific improvement strategy with physics reasoning.