Grade 12 · Week 03 · Wednesday · Day 03

Area Means Displacement

Why does the area under a velocity curve give displacement?

Objective: Calculate signed displacement from velocity–time graph area, including a reversal.

45–60 minutes · Offline-capable · Public practice is not a supervised examination

Read and reason

A rectangle is a displacement

For constant v over an interval, displacement Δx=vΔt. On a velocity–time graph this product is the signed area of a rectangle: m/s multiplied by seconds gives meters. A rectangle below the time axis has negative signed area, representing displacement toward negative x. The signed total may be zero even if the object moved considerable distance.

A trapezoid for constant acceleration

When velocity changes linearly from v₀ to v₁ over duration T, average velocity is (v₀+v₁)/2, so displacement equals T(v₀+v₁)/2. This is the signed area of the trapezoid under the line. It is equivalent to v₀T+½aT² when v₁=v₀+aT. If the line crosses zero, treat positive and negative regions separately when calculating total traveled distance.

Displacement is not distance

Suppose v changes from +4 m/s to −4 m/s over four seconds. Net displacement is (4 + −4)/2×4=0 m. Yet in the first two seconds the velocity is positive and makes a triangle of area +4 m; in the last two seconds the area is −4 m. Traveled distance is 4+4=8 m. One graph produces both answers when the correct sign convention is used.

Distinguish area and slope

Slope of x versus t is velocity. Slope of v versus t is acceleration. Signed area under v versus t is displacement. The units confirm each operation: (m/s)/s = m/s² for slope, and (m/s)×s = m for area. Do not interpret the area under a position–time graph as velocity; that is a different integral with different units.

Vocabulary

signed area
Area under a velocity–time curve counted positive above the time axis and negative below; its sum is displacement. Explain with an original example.
displacement
Signed change of position Δx = xfinal − xinitial; SI unit meter (m). Explain with an original example.
trapezoid
Four-sided area shape; for linear v(t), signed area is duration × (starting velocity + ending velocity)/2. Explain with an original example.
interval
A defined span between starting and ending values of time or another variable. Explain with an original example.
turning point
For constant nonzero a in 1D, an interior instant t=−v0/a where velocity crosses zero and direction reverses. Explain with an original example.

Worked example

v changes linearly from +4 to −4 m/s over 4 s. a=(−4−4)/4=−2 m/s²; displacement=(4−4)/2×4=0 m; distance=8 m due to turning at t=2 s.

Check signs and units. Show each calculation rather than memorizing a number.

Your science journal

Draw a velocity–time graph crossing zero. Calculate positive, negative and absolute area separately, then verify in the shared graph studio.

Journal lives in this browser if localStorage is permitted; export backups to keep it.

Daily Retrieval Practice · 4 Questions

Choose one answer to each question, check, read the feedback, and retry after correcting mistakes. Target 80% open-book mastery; public practice scores do not authenticate official midterms or finals.