SCIENCE · INVESTIGATE AND EXPLAIN

Distinguish a rate from an amount of energy

Energy tells how much is transferred; power tells how quickly. Average power over an interval is transferred energy divided by elapsed time. Instantaneous mechanical power for a force and velocity is their dot product. These quantities agree only for a constant-power interval, not merely because the force is constant.

Learning outcome: At T=2 s compare efficiencies0.4,0.8,1.0. Then keep one efficiency and change duration. Preserve energy and power units and explain why the two comparisons answer different questions.

Starting recap

Choose the system, interval and reference frame before using a conservation statement. Keep vector momentum, scalar energy and transferred power distinct.

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Try

At T=2 s and η=0.5, output energy 16 J requires32 J input. Average output8 W differs from endpoint16 W; average input16 W differs from endpoint32 W.

Open the experiment

Predict

A constant applied force on an accelerating point necessarily gives:

Your prediction and reason

Experiment

At T=2 s compare efficiencies0.4,0.8,1.0. Then keep one efficiency and change duration. Preserve energy and power units and explain why the two comparisons answer different questions.

Separate average power, endpoint power and efficiency

Worked teaching model · synthetic values · Virtual 2 kg point starts from rest on a frictionless line; selected motor provides constant net force 4 N throughout the interval. Constant efficiency relates supplied energy to mechanical output. No physical motor performance or electrical circuit is simulated.

Constant conversion efficiency: 0.8 ratio Powered interval: 3 s

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Mechanical power under constant force

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: P=Fv (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Mechanical power under constant force

P=Fv

time (s)	mechanical output power (W)
0	0
0.075	0.6
0.15	1.2
0.225	1.8
0.3	2.4
0.375	3
0.45	3.6
0.525	4.2
0.6	4.8
0.675	5.4
0.75	6
0.825	6.6
0.9	7.2
0.975	7.8
1.05	8.4
1.125	9
1.2	9.6
1.275	10.2
1.35	10.8
1.425	11.4
1.5	12
1.575	12.6
1.65	13.2
1.725	13.8
1.8	14.4
1.875	15
1.95	15.6
2.025	16.2
2.1	16.8
2.175	17.4
2.25	18
2.325	18.6
2.4	19.2
2.475	19.8
2.55	20.4
2.625	21
2.7	21.6
2.775	22.2
2.85	22.8
2.925	23.4
3	24

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Final speed	6	m/s
Distance during powered interval	9	m
Mechanical output energy	36	J
Supplied input energy	45	J
Other energy increase	9	J
Average mechanical output power	12	W
Endpoint mechanical output power	24	W
Average input power	15	W
Endpoint input power	30	W

Constant force produces increasing power here; average power is not the endpoint value.

From rest with acceleration 2 m/s², v=2t and displacement=t², so output work is 4t². Its average rate over the interval is 4T, while endpoint Fv is 8T. Constant efficiency eta gives input energy Eout/eta; the difference remains in other energy forms, not destroyed energy. Changing duration changes both total energy and power in this stipulated motion.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Constant force produces increasing power here; average power is not the endpoint value.

Observation to attach to a capture

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

Understand

Define the energy or momentum system

Build the algebraic account

A virtual motor exerts constant net force 4 N on a 2 kg point starting from rest. Acceleration is 2 m/s², so after time T the speed is 2T and distance is T². Mechanical output energy is 4T² J. Average output power is 4T W, whereas endpoint power Fv is 8T W, twice the average for this particular ramp from zero speed.

Check an original packet

Conversion efficiency η is the ratio of useful mechanical output to supplied input. Therefore input energy is Eout/η and the difference becomes other energy forms in the stated ledger. At η=0.8 and T=3 s, output is 36 J, input45 J and other energy increase9 J. Average output is 12 W but endpoint output is 24 W; average input is 15 W.

Test the domain and evidence

Changing duration in this model changes both distance and total energy, so it is not a comparison of the same fixed job done at different speeds. To compare efficiencies, hold the mechanical task or interval fixed. The efficiency is stipulated constant here, not a measured universal motor property. No wiring or real power-supply procedure is requested.

Worked original packet

At T=2 s and η=0.5, output energy 16 J requires32 J input. Average output8 W differs from endpoint16 W; average input16 W differs from endpoint32 W.

Bounded investigation

At T=2 s compare efficiencies0.4,0.8,1.0. Then keep one efficiency and change duration. Preserve energy and power units and explain why the two comparisons answer different questions.

Changed-case numerical practice

Try the changed case before feedback

A device delivers useful mechanical energy 30 J in3 s at constant efficiency0.6. Find its average input power in watts.

Numerical result in the requested units

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and limits

Original content checked 30 October 2026 against OpenStax College Physics 2e. All packets and diagrams are synthetic teaching evidence. No physical launches, collisions, spring releases, explosive processes or equipment-safety judgments are requested. Retained self-reports and public numerical feedback do not certify supervised laboratory practice or independent mastery.

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

What measurement would be needed before using a constant efficiency for a real changing-speed motor?

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