SCIENCE · INVESTIGATE AND EXPLAIN

Infer velocity from a position graph with uncertainty

An interval-average velocity is the slope joining two points on a position–time graph: (x₂−x₁)/(t₂−t₁). Instantaneous velocity is the local slope. A graph can sit above zero yet slope downward: positive position and negative velocity are perfectly compatible. Screen angle alone is not a velocity; read axis scales and units.

Learning outcome: Hold acceleration at 1 m/s². Compare half-windows 1, 0.5 and 0.25 s; record the velocity estimate and worst-case bound. Explain why the exact midpoint velocity stays fixed while the measured estimate changes. Repeat with negative acceleration and retain the same error signs.

Starting recap

Use signed quantities, algebraic substitution, graph slopes and areas. State the system, coordinate axes and units before calculating.

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Try

For a=1 m/s² and h=0.5 s, true positions at 1.5 and 2.5 s are 4.125 and 8.125 m. The assigned observations are 4.025 and 8.225 m. Their slope is 4.2 m/s; the true midpoint velocity is 4 m/s. The 0.2 m/s error equals ε/h=0.1/0.5.

Open the experiment

Predict

Will halving the time window always improve this measured slope? Separate the exact quadratic from the observation errors.

Your prediction and reason

Experiment

Hold acceleration at 1 m/s². Compare half-windows 1, 0.5 and 0.25 s; record the velocity estimate and worst-case bound. Explain why the exact midpoint velocity stays fixed while the measured estimate changes. Repeat with negative acceleration and retain the same error signs.

Estimate a local slope with bounded position error

Worked teaching model · synthetic values · Synthetic x=2t+0.5at² m. Each endpoint has an assigned position error bounded by 0.10 m. The displayed left endpoint error is −0.10 m and right is +0.10 m: a deliberately adverse realization, not random noise. Times are exact.

True acceleration: 1 m/s² Half-window around t=2 s: 0.5 s

Reset keeps your prediction, captures and practice history.

Axes rescale with settings; compare numerical values, not screen angles or apparent lengths. Solid/points show the selected model relationship or observations; the dashed comparison is explained below.

Calculated evidence ledger — synthetic teaching model
Quantity	Value	Unit
Left observed position	4.025	m
Right observed position	8.225	m
Secant velocity estimate	4.2	m/s
True midpoint velocity	4	m/s
Worst-case velocity error bound	0.2	m/s

Accessible plot coordinates

Primary plotted points
time (s)	position (m)
0	0
0.25	0.53125
0.5	1.125
0.75	1.78125
1	2.5
1.25	3.28125
1.5	4.125
1.75	5.03125
2	6
2.25	7.03125
2.5	8.125
2.75	9.28125
3	10.5
3.25	11.78125
3.5	13.125
3.75	14.53125
4	16

Dashed comparison coordinates
time (s)	position (m)
1.5	4.025
2.5	8.225

A smaller interval need not give a better experimental velocity estimate: fixed position error is divided by a smaller time interval.

For this quadratic trajectory the exact symmetric secant equals midpoint velocity. The displayed positive bias is entirely the deliberately adverse endpoint error, not curvature bias. Solid: exact trajectory; dashed: observed secant.

A smaller interval need not give a better experimental velocity estimate: fixed position error is divided by a smaller time interval.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

1. Orient: Infer velocity from a position graph with uncertainty

2. Explain the model

For x(t)=2t+0.5at², compare symmetric times 2−h and 2+h. Expanding the squares and subtracting cancels the terms even in h, giving [x(2+h)−x(2−h)]/(2h)=2+2a. This symmetric secant equals the midpoint velocity exactly for this quadratic, though that convenient equality is not true for every curve.

Measured positions include error. If each endpoint can differ from the true position by at most ε, the position difference can err by as much as 2ε. With exact timing, the slope error can therefore reach 2ε/(2h)=ε/h. Decreasing the window may reduce curvature effects but amplifies the influence of fixed coordinate error.

The two error signs in this experiment were deliberately chosen to maximize positive slope error; they are not a random sample. Do not claim a confidence interval from this worst-case bound. To study random precision you would need a justified noise model and repeated independent observations, while calibration bias requires a separate check.

3. Work an example

4. Predict and investigate

Predict first: Will halving the time window always improve this measured slope? Separate the exact quadratic from the observation errors.

5. Transfer and explain

Try a new practice case

Two position readings each have a bounded error of 0.06 m. Times are exact and 0.40 s apart. What is the worst-case average-velocity error bound?

Your numerical result (m/s)

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

How could you reduce error without merely shrinking the interval? Consider calibration, image resolution and fitting several observations.

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: 1.2.B; 1.3.A. 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.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

How could you reduce error without merely shrinking the interval? Consider calibration, image resolution and fitting several observations.

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Distinguish a displacement vector from a route Next: Reconstruct displacement and distance from velocity

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