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Mechanics · Work, energy & power

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Kinetic and potential energy

Kinetic energy ½mv² and gravitational potential energy mgh, the work–energy principle (the work done by the resultant force equals the change in kinetic energy), and conservation of mechanical energy — including motion along a smooth curved path, where only the overall energy change matters.

Mechanics · Work, energy & power

Kinetic and potential energy

Kinetic energy ½mv² and gravitational potential energy mgh, the work–energy principle (the work done by the resultant force equals the change in kinetic energy), and conservation of mechanical energy — including motion along a smooth curved path, where only the overall energy change matters.

Why it works

A moving object has kinetic energy KE=12mv2,\text{KE} = \tfrac12 mv^2, and an object at height hh above some reference level has gravitational potential energy PE=mgh,\text{PE} = mgh, both measured in joules. Throughout Paper 4 take g=10 m s−2g = 10\text{ m s}^{-2}.

The work–energy principle. The work done by the resultant force on an object equals the change in its kinetic energy: Wnet=12mv2−12mu2.W_{\text{net}} = \tfrac12 mv^2 - \tfrac12 mu^2. This is often the quickest route to a final speed, because it links force and distance directly to speed without needing the time or the acceleration.

Conservation of mechanical energy. If the only force doing work is gravity (all other contacts are smooth, so friction and resistance do no work), the total mechanical energy KE+PE\text{KE} + \text{PE} is constant: 12mu2+mgh1=12mv2+mgh2.\tfrac12 mu^2 + mgh_1 = \tfrac12 mv^2 + mgh_2. As an object descends, PE is converted into KE; as it rises, KE is converted back into PE. A tidy consequence for a body released from rest and falling a height hh: all the PE lost becomes KE, so 12mv2=mgh\tfrac12 mv^2 = mgh, giving v=2ghv = \sqrt{2gh} — and the mass cancels.

Curved paths. The power of the energy method is that it does not care about the shape of the path. For a child on a smooth curved slide, you cannot easily use forces (the direction keeps changing), but energy conservation still gives the speed at the bottom from the vertical drop alone — only the overall change in height matters.

When friction acts. Mechanical energy is no longer conserved: some is spent doing work against resistance. Then account for it explicitly, energy at start=energy at end+work done against resistance,\text{energy at start} = \text{energy at end} + \text{work done against resistance}, which is just the work–energy principle written out with the resistive force included.