The Trolley Down the Ramp
A practical group investigates acceleration using a trolley released from rest on a ramp of length 1.20 m. They time five runs with a stopwatch, obtaining 1.42, 1.38, 1.55, 1.40, and 1.39 s. One student computes acceleration from s = ½at² using each time and averages the results. Another computes a single acceleration from the mean time. A third notices that the 1.55 s run was the one where the trolley scraped the side of the ramp. The ramp angle is set by a 10 cm block under a 1.00 m length of track. The group must decide how to process the data and what uncertainty to quote.
1. Compare the two data-processing methods and recommend one, justifying your choice.
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Using each time to get a = 2s/t² and then averaging gives: for 1.42 s, a ≈ 1.19 m/s²; 1.38 s, 1.26; 1.55 s, 1.00; 1.40 s, 1.22; 1.39 s, 1.24. The mean is about 1.18 m/s², but the 1.55 s value drags it down and the averaging of nonlinear results biases the outcome. Using the mean time (1.428 s) gives a = 2 × 1.20 / 1.428² ≈ 2.40/2.039 ≈ 1.18 m/s². Both give similar values here, but the mean-time method is cleaner because acceleration depends on t², so averaging accelerations weights short times disproportionately. Recommendation: compute a single acceleration from the mean of the valid times, after excluding the 1.55 s run as an identified anomaly (trolley scraping). That gives mean time 1.398 s and a ≈ 2.40/1.954 ≈ 1.23 m/s².
2. Estimate the acceleration expected from the ramp geometry and comment on the agreement with the experimental value.
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The ramp rises 10 cm over 1.00 m, so sin θ ≈ 0.10 and θ ≈ 5.7°. For a frictionless trolley, a = g sin θ ≈ 9.81 × 0.10 ≈ 0.98 m/s². The measured value of about 1.23 m/s² is higher than this prediction, which is suspicious because friction should reduce acceleration, not increase it. Possible explanations: the block height or track length was measured differently than assumed, the trolley was pushed slightly at release, or timing started late. The judgement is that the discrepancy (about 25%) is too large to ignore and the group should recheck the geometry and release procedure before trusting the result.