Integrate power without confusing one interaction with net work

Differentiate energy transfer in time

For a selected force, instantaneous power is dW/dt=F·v. Average power over a finite duration is total selected work divided by that duration. These quantities coincide only under special histories, not simply because the same units are used. Power can be negative when the interaction removes energy.

Use the actual vector velocity

The supplied path is x=t²,y=t in SI numerical form, so velocity is (2t,1). The selected force is (at,2) N. Their dot product gives P(t)=2at²+2 W. Multiplying force magnitude by speed would miss the angle and lose sign, especially when a is negative.

Integrate the history

Over 0≤t≤T, selected work is 2aT³/3+2T. Dividing by T gives average power 2aT²/3+2, while the endpoint instantaneous power is 2aT²+2. For a=1,T=2, work 28/3 J, average 14/3 W and endpoint 10 W are three distinct quantities.

Retain the interaction boundary

The path is prescribed and other interactions may be needed to enforce it. The displayed force is selected, not declared net. Therefore its power does not alone determine the derivative of the object’s entire kinetic energy. A negative selected work can coexist with increasing K if other interactions supply more energy.

Challenge a constant-power shortcut

For an ideal net constant power P applied to a positive-speed mass, K=K 0+Pt and speed=sqrt(2K/m). The required along-motion force is P/v. Extending nonzero constant power down to exactly zero speed would require unbounded force in this idealization. A real start needs a different force-limited phase; the algebra is not an actuator specification.

Worked original case

a=1,T=2 gives W=28/3 J, average 14/3 W, endpoint 10 W. With a=−1, power crosses zero at t=1 and becomes negative afterward, while prior transferred work remains part of the interval integral.

Investigation

Compare a=1 and−1 at T=2. Derive each power curve, find any sign change and integrate signed areas. Explain what extra evidence would be needed to infer net kinetic-energy change.

Public numerical transfer practice

Try a new practice case

For force(3,4) N and velocity(2,−1) m/s, what is instantaneous selected power in watts?

Numerical result in the requested units Check this practice result

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Original teaching, values and calculations, checked 30 October 2026 against the official calculus-based Mechanics framework and OpenStax work and kinetic energy. The numerical packets are supplied or generated, not measured apparatus records. No physical launch, machinery, loading or field intervention is requested. Public feedback and virtual captures do not establish independent mastery or supervised laboratory participation.