Measurement and Model Testing

In the previous lesson, converting 54 km/h to 15 m/s preserved the supplied speed. But how closely does a supplied number describe the actual motion? That question takes us from calculating with quantities to measuring them. Suppose two timers report 0.75 s and 0.76 s for the same cart trip. To decide whether that difference matters, we need to know how the trip was timed and how much the readings could vary.

After Units, Dimensions, and Estimation, you can keep quantities consistent. This lesson adds a way to describe their limitations: distinguish resolution, repeatability, and bias, and design a fair model test. It uses arithmetic and intervals, without calculus or formal statistical inference.

Define the measurement before reporting its number

A timing measurement needs a start event and an end event. “Time of the cart” is incomplete. “Time from the center passing the first mark to its passing the second” defines a reproducible procedure. Record the selected system and frame from Physical Models, Systems, and Frames.

Start with the instrument. Calibration means comparing its readings with a known reference to investigate and, where appropriate, correct a consistent error. On an ideal, calibrated ruler marked every 1 mm, rounding a clear reading to the nearest mark changes it by at most 0.5 mm. This tells us the limitation due to rounding alone. Alignment with the object, the width of its edge, and a shifted ruler zero can add further error. Likewise, a timer may display hundredths of a second while the person operating it starts late by much more than 0.01 s.

Repeatability and accuracy answer different questions

Repeatability describes how closely repeated results agree under the same conditions. Accuracy concerns closeness to the value of the quantity being measured. Closely grouped readings can all share a systematic offset.

The timing example separates two kinds of measurement limitation. A random contribution varies between trials and produces scatter among the readings. A systematic contribution shifts results consistently under the procedure; the timer’s added 0.12 s is an example. Such a consistent shift is called bias. Repeating measurements helps us describe scatter, while calibration, alignment checks, or a second measurement method help us investigate bias. We need both kinds of checks to judge a result.

Design a test before seeing its result

For a falling-object investigation, choose a safe, small drop height and a clear arrival event. An air-free model predicts equal arrival times for two objects released from rest at the same height under the same gravitational conditions; later units derive that prediction. Treat it here as a supplied model prediction to test, not a result derived in this lesson.

Keep release height and method matched. Use repeated trials, estimate timing limitations, and check for release delays. Decide in advance what difference would be distinguishable with the justified uncertainty. The separate uncertainty lesson develops numerical comparison criteria. Here, focus on defining the arrival event, keeping the conditions matched, and checking the timing method before drawing a conclusion.

Check your understanding

1. A timer displays 0.01 s increments but consistently starts 0.15 s late. Does its display resolution bound the full error?
2. The five timing trials have a mean recorded interval of 0.75 s. An independent calibration shows that the timer adds 0.12 s to each reading. What is the corrected mean, and what happens to the spread of the readings?
3. Why can closely grouped repeated readings still give an inaccurate result? Name one check that could reveal the problem.
4. Design a repeatable comparison of two release methods for a falling ball. Specify matched conditions, a timing event, a way to check bias, and a decision criterion established before measurement.

Reveal suggested answers

Suggested answers

1. No. Start delay is a separate contribution; resolution alone does not capture it.
2. The corrected mean is $0.75\,\mathrm{s}-0.12\,\mathrm{s}=0.63\,\mathrm{s}$. Subtracting the same offset from every reading leaves their spread unchanged.
3. All the readings may share a systematic error, such as a shifted timer zero. Compare the instrument with a known reference or use an independent measurement method to investigate it.
4. One suitable design uses the same ball, height, arrival detector, and frame, changing only the release method. Repeat and alternate methods, use video or a second timing method to check start delay, and choose a distinguishability threshold based on justified timing bounds. Alternation reduces the risk that a slow drift is mistaken for a method difference. Repeated scatter alone does not justify the full bounds.

Connections and references

Continue with Significant Figures and Reporting Measurements, then Uncertainty Bounds and Comparing Results. Use the Model and Units Checklist to record those decisions. These habits support motion graphs, force measurements, spring experiments, and the final mechanics investigation. The next unit begins with position, displacement, and distance; no additional statistics or calculus is required to enter it.

Reference basis: Serway and Vuille, College Physics, 9th edition, §§1.4, 1.6, 1.9; Verma, Concepts of Physics, part 1, §§2.12–2.14; Young and Adams, College Physics, 11th edition, chapter 1, especially §§1.2–1.6. These sources inform the measurement and reporting discussion; no confidence claim is inferred from sample range or standard deviation. All practice data are invented and all numerical examples are original.