Trolley Down a Ramp: Reconciling Two Measurements of Acceleration
A Grade 11 lab group releases a frictionless dynamics trolley from rest at the top of a 1.20 m ramp inclined at 8.0° to the horizontal. A motion sensor records position every 0.02 s and the software fits a parabola, returning a = 1.05 m/s² with a stated uncertainty of ±0.04 m/s². Separately, students time the run with a stopwatch three times: 1.52 s, 1.60 s, and 1.48 s, and compute a = 2s/t² from the full 1.20 m. The accepted value from g·sinθ (g = 9.81 m/s²) is 1.37 m/s². The teacher must decide what to tell the group and how to frame the discrepancy for the class.
1. Compute the stopwatch-based acceleration and its spread, then judge whether each method agrees with the theoretical value within its own uncertainty. Which measurement is more trustworthy, and why?
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Stopwatch method
Mean time: t̄ = (1.52 + 1.60 + 1.48)/3 = 1.533 s.
a = 2s/t² = 2(1.20)/(1.533)² = 2.40/2.350 = 1.02 m/s².
Spread: range 1.48–1.60 s. Using the extremes:
- t = 1.48 s → a = 2.40/2.190 = 1.10 m/s²
- t = 1.60 s → a = 2.40/2.560 = 0.94 m/s²
So the stopwatch result is roughly 1.02 ± 0.08 m/s² from timing alone - and human reaction time (~0.2 s) adds a systematic component that this spread does not capture.
Comparison with theory (1.37 m/s²)
- Motion sensor: 1.05 ± 0.04 → discrepancy 0.32 m/s² = 8σ. Does not agree.
- Stopwatch: 1.02 ± 0.08 → discrepancy 0.35 m/s² ≈ 4σ. Does not agree.
Judgement: the motion sensor is more trustworthy because it samples continuously, its uncertainty is quantified, and it is not limited by reaction time. But both methods sit well below theory, so the problem is almost certainly systematic, not statistical. The stopwatch's apparent agreement with the sensor is coincidental - two flawed measurements agreeing does not validate either.
Next diagnostic step: check whether the ramp angle was measured correctly, whether the trolley actually started from rest, and whether the track was level across its width (a twisted track reduces the effective incline).
2. The group insists their data "prove" g is smaller in the lab. Write the feedback you would give, naming the specific reasoning error and the concrete next step.
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The reasoning error is treating a single consistent discrepancy as confirmation rather than as a signal to look for a systematic effect. Two independent methods agreeing is only meaningful if their systematic errors are independent - here both share the same ramp, the same 1.20 m distance, and the same assumption that the trolley starts from rest.
Feedback to give:
- Name what the data support: "Your data are precise and internally consistent. They do not yet tell us why they differ from g·sinθ."
- Distinguish precision from accuracy: the sensor's ±0.04 m/s² is a precision claim; an 8σ offset is an accuracy problem.
- Propose a falsifiable test: repeat with the ramp at 12° and 16°. If the measured acceleration scales as sinθ, the model is right and the angle or distance calibration is wrong. If the ratio stays constant, the model itself is being misapplied.
- Check the assumed inputs directly: re-measure the incline with a digital level, verify the release is truly from rest, and confirm the sensor's position calibration against a metre rule.
Frame it as: "A result that disagrees with theory is more interesting than one that agrees - but only if you chase down the reason."