AS Daily A Level Physics question
A student estimates the acceleration a of a trolley down a shallow ramp from measurements of distance s and time t using a = 2s/t^2. They record s = 0.80 m with a metre rule (resolution 1 mm) and t ≈ 1.2 s with a hand-held stopwatch (about ±0.20 s per timing). They want to cut the percentage uncertainty in a to roughly half without changing the type of equipment. Which change should they make? Assume the true acceleration does not change.
Answer
The correct answer is D.
Correct: D — Increase s to 3.20 m so that t approximately doubles. Because a ∝ s/t^2, the fractional uncertainty combines as Δa/a ≈ Δs/s + 2Δt/t. Initially: Δs/s ≈ 0.001/0.80 = 0.125% and 2Δt/t ≈ 2×0.20/1.2 ≈ 33.3%, so ≈ 33.5% total. Quadrupling s doubles t (for the same a), giving Δs/s ≈ 0.001/3.20 = 0.031% and 2Δt/t ≈ 2×0.20/2.4 ≈ 16.7%, so ≈ 16.7% overall — about half. A Doubling mass does not change s, t or their uncertainties here, so the percentage uncertainty in a is essentially unchanged. B Halving s reduces t by √2, so the timing term grows: 2Δt/t ≈ 2×0.20/0.85 ≈ 47% and Δs/s doubles to 0.25%, making the overall uncertainty worse. C Averaging four repeats does not reduce a consistent stopwatch offset (systematic); if the same offset is assumed each time, the dominant 2Δt/t term is not improved, so the percentage uncertainty is not halved.