Physics • Year 11 • Module 2 • Lesson 5
Acceleration and Graphical Analysis
Build HSC Band 5–6 extended-response technique on deriving Newton’s Second Law from graphical data, evaluating experimental conclusions, and designing controlled investigations.
1. Data + scenario: deriving F = ma from a trolley experiment (Band 5–6)
8 marks Band 5–6
Scenario. A student investigates Newton’s Second Law using a trolley on a frictionless track. She keeps the mass constant at 5.0 kg and applies increasing net forces using hanging masses over a pulley, measuring the resulting acceleration with a motion sensor. The table below shows her results, and the F vs a graph is plotted alongside.
| Trial | Net Force Fnet (N) | Acceleration a (m s−2) |
|---|---|---|
| 1 | 4.0 | 0.80 |
| 2 | 8.0 | 1.62 |
| 3 | 12.0 | 2.40 |
| 4 | 16.0 | 3.18 |
| 5 | 20.0 | 4.05 |
Data from Activity 01 in the lesson. Trolley mass = 5.0 kg throughout.
Q1. Analyse and evaluate the data above to determine whether the results support Newton’s Second Law, and assess the quality of the experimental method. In your response you must:
- Calculate the gradient using the two labelled triangle points on the line of best fit. Show rise/run working and include units.
- Interpret the gradient in NESA-standard language, linking it to Newton’s Second Law in the form F = ma (y = mx).
- Use the F/a ratio across all five trials to assess the precision of the results. State whether the data supports F ∝ a.
- Explain what the straight line through the origin indicates about the relationship between F and a at constant mass.
- Identify one source of systematic or random error in the experiment and suggest an improvement to reduce it.
2. Experimental design — testing whether a heavier trolley accelerates less (Band 5–6)
7 marks Band 5–6
Research question. A student claims that “if you keep the force the same and increase the mass, the acceleration halves each time you double the mass.” Design a controlled investigation to test this claim experimentally.
Constraints: You have access to a frictionless (air-track) cart, hanging masses (0.1–1.0 kg), extra mass blocks to add to the cart (1.0, 2.0, 4.0 kg), a motion sensor, a light gate, and a metre ruler. The investigation must be completable in one 50-minute lesson.
Q2. Design the investigation and present it in the format below.
- State your hypothesis as a testable prediction including the independent variable and expected result.
- Identify the independent variable, dependent variable, and at least two controlled variables with reasons for controlling them.
- Describe the procedure in at least four numbered steps, including how you will ensure the force remains constant throughout each mass variation.
- Describe how you will process the data (what graph you will draw, what the gradient should be, and how you will confirm the hypothesis).
- State one result that would falsify your hypothesis and one limitation of the design.
Q1 — Sample Band 6 response (8 marks), annotated
Gradient calculation (marks 1–2): Using the labelled triangle points: ΔF = 15 N (rise); Δa = 3 m s−2 (run) [1]. Gradient = ΔF / Δa = 15 / 3 = 5.0 kg. Units: N ÷ m s−2 = kg [1].
NESA interpretation (mark 3): The gradient of the F vs a graph represents the mass of the trolley (5.0 kg), because Fnet = ma can be rearranged to F = m × a, which has the form y = mx where the gradient m equals mass [1].
F/a ratio assessment (mark 4): F/a ratios: 4.0/0.80 = 5.0; 8.0/1.62 ≈ 4.9; 12.0/2.40 = 5.0; 16.0/3.18 ≈ 5.0; 20.0/4.05 ≈ 4.9. All ratios ≈ 5.0 kg (variation <2%), which closely matches the known trolley mass. This confirms F ∝ a at constant mass, providing quantitative support for Newton’s Second Law [1].
Straight line through origin (mark 5): The straight line through the origin confirms direct proportionality between net force and acceleration at constant mass. If the line did not pass through the origin, it would indicate a systematic error such as a constant friction force (y-intercept > 0) or a calibration offset [1].
Error and improvement (marks 6–8): One systematic error is residual friction in the cart wheels or pulley, which means the measured acceleration is consistently lower than the theoretical value. This shifts all data points below the expected line and may produce a positive y-intercept [1]. One improvement: use a frictionless air-track and a lightweight, frictionless pulley to eliminate friction [1]. Additionally, conducting three repeat trials for each force level and averaging the acceleration would reduce the impact of random timing errors from the motion sensor [1].
Marking criteria summary (8 marks): 1 = gradient calculated correctly using points on line of best fit with rise/run shown; 1 = correct units (kg) stated and confirmed by unit analysis; 1 = NESA-standard gradient interpretation naming mass and linking to F = m × a in form y = mx; 1 = F/a ratio analysis with all five values and correct conclusion; 1 = straight line through origin interpreted correctly (direct proportionality); 1 = one valid source of error identified with physical explanation; 1 = one specific improvement; 1 = uses precise physics terminology throughout (Newton’s Second Law, direct proportionality, line of best fit, gradient, systematic error).
Q2 — Sample Band 6 response (7 marks), annotated
Hypothesis: If mass is doubled while net force remains constant, the acceleration will halve. Independent variable: total mass of cart (1.0, 2.0, 4.0 kg). Dependent variable: acceleration of the cart (m s−2), measured via light gate or motion sensor. [1 — hypothesis with IV and DV]
Controlled variables: (1) Applied force — kept constant by using the same hanging mass (e.g. 0.5 kg) throughout; if force varied, the change in acceleration could not be attributed solely to mass change. (2) Track surface — same air-track used throughout so friction conditions are identical. [1]
Procedure: (1) Set up the air-track horizontally; attach a 0.5 kg hanging mass via a string over a pulley to the cart. Measure the net force F = 0.5 × 9.8 = 4.9 N. (2) Load the cart with the 1.0 kg block; total cart mass = 1.0 kg + cart mass. Release from rest and record acceleration from the motion sensor over 1.0 m. (3) Repeat with 2.0 kg, then 4.0 kg mass blocks on the cart. Conduct three trials per mass; record mean acceleration. (4) Calculate F/a for each mass; plot acceleration (y-axis) vs 1/m (x-axis) and draw line of best fit. [1 — four clear steps including method to keep force constant]
Data processing: Plot a vs 1/m; if a ∝ 1/m (at constant F), the graph will be a straight line through the origin with gradient = F. Calculate predicted acceleration for each mass from a = F/m and compare with measured values. [1]
Falsification: If doubling the mass does not halve the acceleration (e.g. mass doubles but acceleration drops to only 60% rather than 50% of the original), or if the graph of a vs 1/m is curved rather than straight, the hypothesis would be falsified [1].
Limitation: The hanging mass and cart mass are not negligible compared to each other; the accelerating system includes the hanging mass, so total mass = cart mass + hanging mass. If the student uses only the cart mass in calculations, the gradient will not equal F exactly [1].
Marking criteria summary (7 marks): 1 = testable hypothesis with IV (mass) and DV (acceleration) stated; 1 = two controlled variables with physical reasons; 1 = four-step procedure including method for keeping force constant (same hanging mass); 1 = data processing described (graph type, what gradient means, how to confirm hypothesis); 1 = valid falsification criterion; 1 = one specific, correctly reasoned limitation; 1 = uses precise terminology (independent/dependent/controlled variable, direct proportionality, acceleration, systematic error).