Reconstruct displacement and distance from velocity

1. Orient: Reconstruct displacement and distance from velocity

A velocity–time graph answers two different questions. Its slope gives acceleration, with units (m/s)/s. Its signed area gives displacement, with units (m/s)×s. A point on the vertical axis is velocity itself, not displacement or acceleration. Before calculating, write the quantity and the units you need.

2. Explain the model

For a straight velocity graph, the signed area can be found from trapezoids or triangles. If the graph crosses zero, split at the crossing. Area below zero subtracts from displacement because the object moves along the negative axis. The magnitude of that same area adds to distance because distance never cancels a return journey.

If v=v₀+at, a possible zero occurs at t=−v₀/a when a is nonzero. Count a reversal only if that time lies strictly inside the observed interval and the sign changes. A zero at the final instant is not evidence that the object has already returned; a zero velocity with zero acceleration can persist indefinitely.

Average velocity divides signed displacement by elapsed time. Average speed divides distance by elapsed time. They agree in magnitude only when the object does not reverse direction over the interval. Negative acceleration can increase speed if the object is already moving negatively.

3. Work an example

For v₀=3 m/s and a=−1 m/s², the graph crosses zero at 3 s. From 0 to 3 s the area is +4.5 m. From 3 to 4 s it is −0.5 m. Displacement is 4 m, distance is 5 m, average velocity is 1 m/s and average speed is 1.25 m/s.

4. Predict and investigate

Predict first: For a graph with equal positive and negative triangular areas, can displacement vanish while distance remains positive?

Compare (v₀,a)=(3,−1), (4,−1), and (−3,1). Calculate each area before checking the table. Locate any interior reversal and explain the boundary case where velocity reaches zero exactly at 4 s. Record both displacement and distance. Use the lesson experiment controls and retain the settings with your evidence.

5. Transfer and explain

Try a new practice case

An object has v=4−2t m/s from t=0 to t=3 s. Find total distance traveled.

Your numerical result (m) 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

Why is |average velocity| insufficient to reconstruct distance? Include a counterexample with an actual direction change.

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.