Test whether agreement between methods is independent evidence

1. Frame the question

Two methods that agree can still share the same error. A freefall timing method infers gravity fromg=2h/t²; a small-angle pendulum infersg=4π²L/T². Their mechanisms differ, but if both use the same miscalibrated clock, their resulting estimates are not fully independent. The purpose of this capstone is to design evidence that could expose such a shared cause.

2. Build the mechanism

The supplied generator has trueg=10 m/s², exact heights and lengths, negligible drag and ideal small-angle motion. A clock scale c multiplies every true elapsed time. Freefall alone also has an added endpoint timing lagd. For h=0.80 m, true fall time is0.40 s and the observed time is0.40c+d. For L=1 m, the observed pendulum period is c×2π/√10. These are prescribed synthetic biases, not observed instrument calibration results.

If d=0, both inferred gravities equal10/c². They agree exactly even when c differs from one. Ifd is nonzero, the fall result shifts separately. A disagreement can reveal inconsistency, but it does not uniquely identify which method failed. Independent clock calibration, independent length checks, changed timing procedures and repeated observations address different alternatives.

A useful report states its question, variables, repeated measurements, controls, graphical analysis, uncertainty and falsification criterion before viewing a confirming result. The time-squared versus height graph is linear through the origin for zero lag under the model; an added timing lag changes its form. Precision among repeated readings cannot exclude a common calibration bias. The displayed generator truth is available for teaching, whereas a real investigator would need independent reference evidence.

3. Work an example

With clock scale1.05 and no added fall lag, fall time is0.42 s and both methods infer10/1.05²≈9.0703 m/s². Their difference iszero, but both differ from the generator truth. With an added fall lag0.05 s, fall time is0.47 s and fall gravity becomes1.6/0.47²≈7.2431 m/s²; the pendulum estimate remains9.0703. This contrast challenges a fall-specific assumption but does not itself certify the clock.

4. Predict and investigate

Can agreement between two different physical methods prove that their shared timing scale is correct?

Compare c=1 and1.05 with lagzero, then introduce lag0.05 at fixedc. Save both method estimates and the multiheight fall packet. Propose a genuinely independent calibration and predict how its evidence would differ under clock-scale versus fall-lag explanations. Specify how length error could confound the comparison.

5. Transfer to a new case

Try a new practice case

A shared clock reads1.10 times true elapsed time. True gravity in a supplied generator is9.68 m/s². With no other error, find either method’s inferred gravity.

Your numerical result (m/s²) Check number

Reveal worked check (practice, not mastery)

This extra practice is not saved and does not award mastery. Record your first answer, explanation and whether you revealed help in the lesson notebook. Use the lesson’s saved question separately.

6. Audit the evidence

Which claims in your portfolio are exact within a stipulated model, supported by supplied observations, or still awaiting real supervised measurement and independent assessment?

Retain the first prediction, contrasting settings, units, derivation, uncertainty assumptions and a changed-case explanation in the course workspace. Public worked answers do not constitute secure independent assessment. Supplied-data captures are not hands-on laboratory time or observed instrument competence. A practical proposal requires qualified local review before any physical execution.

Sources

Original synthesis cases and code, checked30 October2026 against the current AP Physics1 CED, OpenStax work and energy, inelastic collisions, and pendulum assumptions. The common-calibration counterexample and numerical packets are original, not official exam items.