Use a force-position graph to calculate work

1. Orient: Use a force-position graph to calculate work

When force changes with position, multiply force by displacement only after accounting for that change. Partition the path into small intervals and add force component times interval length. On a force-versus-position graph, this is signed area. Straight segments allow exact triangle and trapezoid geometry without calculus.

2. Explain and derive

For a linear force change from F₀ to F₁ across distance L, the average force over position is(F₀+F₁)/2 and work is(F₀+F₁)L/2. This is a position-average, not automatically a time-average. A graph of force against time instead gives impulse, not work; its horizontal axis changes the physical meaning of area.

If the graph crosses zero, split the area into positive and negative parts. Their signed sum determines net work. Adding their magnitudes answers a different bookkeeping question and cannot be substituted into ΔK. A steep force-position slope may be associated with stiffness, but slope is not the work.

The current object starts with 50 J kinetic energy so every supported force history can traverse the full 4 m interval without exhausting it. In a different case, if cumulative negative work reduced K to zero before the endpoint, continuing along the same direction would require checking the motion and force model. A negative computed kinetic energy signals an inaccessible assumption, not a valid result.

3. Work an example

A force changes linearly from 8 N at x=0 to−4 N at x=4 m. Trapezoidal area is(8−4)×4/2=8 J. It crosses zero at x=8/3 m: positive triangular area is 32/3 J and negative-area magnitude 8/3 J. Their signed sum is 8 J, so final kinetic energy is 58 J.

4. Predict and investigate

Predict first: If the endpoint forces are equal and opposite, can the force be nonzero along most of the path yet do zero net work?

Compare endpoint pairs(8,−4),(4,−4) and(−8,−4) N. Sketch the line and signed areas first. Use both geometric splitting and the trapezoid expression. Check that the final kinetic energy equals 50 J plus the calculated work. Use the lesson controls and retain the settings with your evidence.

5. Transfer and explain

Try a new practice case

Net force varies linearly from−2 N to−6 N over 3 m. Find net work.

Your numerical result (J) 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

What extra information would be needed to calculate work from a force-versus-time record? Why is multiplying its area by distance not correct?

Retain a first attempt, signed ledger, units, system boundary, assumptions and a changed-case prediction. Public worked checks are practice, not independent mastery evidence. Synthetic outputs are not physical measurements.

Framework and source notes

Original teaching prose and numerical cases. College Board AP Physics 1 CED, Unit 3 topic mapping: 3.2. Checked 30 October 2026; this is a content cross-reference, not AP authorization.