Problem — Manual stopwatch versus photogates

Problem

A classmate measures the fall time of an object with a manual stopwatch and always obtains values between $0{,}45$ and $0{,}55$ s. A second group, using photogates, obtains $0{,}502 \pm 0{,}003$ s. Discuss qualitatively the difference between random error and systematic error in the two cases, and how to reduce each.

Solution

Random error. This is the error that varies unpredictably from one measurement to the next, scattering the values around the mean. In the case of the manual stopwatch, the dominant cause is the operator’s reaction time (of the order of $0{,}2$ s), which changes each time depending on how early or late the button is pressed. This is exactly what makes the results oscillate between $0{,}45$ and $0{,}55$ s: a spread of roughly $\pm 0{,}05$ s. Photogates, on the other hand, start and stop the count automatically as the object passes, eliminating human reaction time: the spread drops to $\pm 0{,}003$ s. This is why the photogate measurement has a much smaller random error.

How is random error reduced? By repeating the measurement many times and taking the average: the uncertainty on the mean decreases as $1/\sqrt{N}$, where $N$ is the number of measurements. By averaging enough manual-stopwatch measurements the result can be tightened, but it is still preferable to use an instrument (photogates) that cuts down the random error at its source.

Systematic error. This is an error that always repeats in the same direction and by the same amount, for example a mis-calibrated instrument (badly positioned photogates, a stopwatch that starts with a fixed delay). Unlike random error, systematic error cannot be reduced by averaging many measurements: the mean would always remain shifted by the same amount. To correct it, the cause must be addressed, that is, the instrument must be calibrated against a known reference, or the measurement corrected afterwards once the deviation has been estimated.

$\ev{\text{Casuale} \to \text{media su } N \text{ misure } (\propto 1/\sqrt{N}); \quad \text{Sistematico} \to \text{taratura}}$

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Topics: Experimental method Concepts: Significant figures and uncertainty