Work and energy

Objective. Use work-energy conservation with constraints.

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Learn: the key idea

Work and energy methods use the work-energy theorem: the work done by the net force equals the change in kinetic energy, W = ΔKE = ½mv² − ½mu². For conservative forces like gravity, total mechanical energy (kinetic plus potential, mgh) is conserved, allowing speed to be found at any height without working through the full equations of motion. A ball dropped from height h has speed v = √(2gh) at the ground, found by setting mgh = ½mv². Constraints, such as a bead on a wire, let you equate energy at two points directly, often avoiding force analysis entirely.

Worked example

A 2 kg object falls from rest through height 5 m with no resistance; using work and energy, find its speed on landing (g = 10 m/s²).

1 $Energy conservation: mgh = ½mv²$
2 $Cancel m: gh = ½v²$
3 $Substitute: 10(5) = ½v², so v² = 100$
4 $Take square root: v = 10$

Answer: 10 m/s

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Common mistake. Applying conservation of mechanical energy when friction or air resistance does work on the system; energy is only conserved for conservative forces, so non-conservative work must be included separately.

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