Work, Energy & Power — Quick Summary

Work

Working transfers energy in an organised way (particles move together); heating transfers it in a disordered way (temperature difference). Work is:

Only the force component in the direction of motion does work — a force at 90° to displacement does zero work. Units: N m, or joules.

Stretching objects (Hooke's law)

Since force grows steadily from zero, work done is the average force × extension:

Force–displacement graphs

Area under a force–displacement graph = work done.

For a straight line (e.g. Hooke's law), area = a simple triangle or rectangle. For a curved line: split into strips and sum, or integrate — W=∫F ds.

Deriving the energy equations

Both familiar equations come directly from W=Fs: substituting F=mv/t and s=½vt (from SUVAT, u=0) gives W=½mv²; substituting F=mg and s=Δh gives ΔE_p=mgΔh — it's the change in height that matters, not absolute height.

Conservation of energy

Energy is never created or destroyed, only transferred — the total in a closed system stays constant. With resistive forces present:

Power & efficiency

One watt = one joule transferred per second. P=Fv is easy to overlook: the same resistive force at different speeds needs very different power (think wading through water quickly vs. slowly).

Worked example: cyclist on a hill

At steady speed, driving force = weight component along the slope, mg sinθ.

At steady speed (equilibrium), driving force = mg sinθ. Combined with P=Fv, measured power and speed give θ.

Zig-zag path (same height gained, longer distance, same speed): takes longer, so the same energy is transferred over more time → lower power. Equivalently, the shallower effective angle means a smaller mg sinθ, so a smaller F and (via P=Fv) a smaller P — both routes agree.

Curved graph → acceleration isn't constant; use a tangent's gradient, not SUVAT.

Freewheeling downhill: GPE converts to KE, but resistive forces grow with speed, taking a larger share of the GPE lost per second — so acceleration decreases. Eventually resistive forces balance the weight component down the slope, and the cyclist reaches terminal velocity (the graph levels off).