Infer a parameter and challenge it with unused evidence

1. Orient: Infer a parameter and challenge it with unused evidence

Parameter fitting is a claim about a model, not just a way to draw a line. For a release from rest at a known zero position with constant acceleration, x=0.5at². Plotting x against t² makes a straight-line model with slope k=a/2. A nonzero initial velocity or an incorrect origin would require extra terms rather than silently altering acceleration.

2. Explain the model

To choose a through-origin slope from several observations, minimize the sum of squared vertical residuals. The resulting slope is k=Σ(tᵢ²xᵢ)/Σ(tᵢ⁴). You can use the formula without calculus: each candidate predicts k times the horizontal coordinate, and the selected value balances the signed weighted residuals. Acceleration is twice this slope, not the slope itself.

The supplied positions are original synthetic values with an explicitly bounded error pattern. For exact times and |position error|≤ε, the worst-case slope error is at most εΣ(tᵢ²)/Σ(tᵢ⁴). Double it for the acceleration bound. This conservative calculation does not assume independent random errors and does not create a statistical confidence level.

A held-out reading was not used to choose the slope. Its residual tests a prediction rather than merely repeating the fit. Combine the predicted-position bound with the held-out measurement bound before judging compatibility. Passing the check means the data do not reject this model under these bounds; it does not prove that acceleration is constant or that the instrument has no common calibration bias.

3. Work an example

Using just x=1.02 m at t=1 s and x=3.98 m at t=2 s gives k=(1×1.02+4×3.98)/(1+16)=16.94/17≈0.99647 m/s², so a≈1.99294 m/s². If each position error is bounded by 0.03 m, the acceleration bound is 2×0.03×5/17≈0.01765 m/s². These two readings are an illustrative worked case, separate from the interactive packet.

4. Predict and investigate

Predict first: Will using more sampled points guarantee protection against one shared calibration-scale error? Explain before fitting.

Compare sampling intervals 1,0.5 and 0.25 s at the same position bound. Record the number of selected observations, acceleration estimate, bound and held-out residual. Then change the error bound. Explain why passing a held-out check is conditional and why correlated calibration error is not eliminated by more points. Use the lesson experiment controls and retain the settings with your evidence.

5. Transfer and explain

Try a new practice case

A rest-release dataset at a known zero position gives a fitted slope of 0.85 m/s² on an x-versus-t² graph. Infer acceleration.

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. Reflect and retain evidence

Design a second method for estimating acceleration from the same footage, such as velocity changes over separated windows. Name a calibration check that is independent of both fits. Real practical competence requires supervised evidence; this supplied-data exercise cannot certify it.

Record a claim, numerical or graphical evidence, assumptions, a limitation and a changed-case prediction. A supplied synthetic investigation is not evidence that you performed a physical laboratory.

Framework and source notes

Original teaching prose, cases and synthetic datasets. Framework mapping: 1.3.A; science practices 1, 2, 3. College Board AP Physics 1 CED, Unit 1, checked 30 October 2026. This mapping is a content cross-reference, not AP authorization or a claim of full course completion.