Work, energy and power

work energy theoremconservation of energy formulaskinetic and potential energypower formulas

Work, the two stores of mechanical energy, the theorem that connects them to speed, and the two ways of writing power.

Work (W = Fd cos θ)

W = F d \cos\theta

Work done by a constant force acting at an angle to the displacement.

Gravitational Potential Energy (U = mgh)

U = m g h

Energy stored by raising a mass to height h near Earth's surface, with g = 9.80665 m/s².

Elastic Potential Energy

U = \tfrac{1}{2} k x^{2}

Energy stored in an ideal spring displaced x from its rest length.

Work–Energy Theorem

W = \tfrac{1}{2} m \left(v^{2} - v_0^{2}\right)

Net work done on an object equals its change in kinetic energy, linking force and distance to a change in speed.

Power (P = W/t)

P = \frac{W}{t}

Average power as work or energy delivered per unit time.

Power from Force and Velocity (P = Fv)

P = F v

Instantaneous power delivered by a force parallel to the velocity.

Machine Efficiency

\eta = \frac{W_{out}}{W_{in}}

Efficiency of a machine as useful work out divided by work in, with the shortfall lost to friction, heat, and noise.

How they fit together

Work is the transfer of energy by a force, so every formula here is a different account of the same joule. The work–energy theorem is the hinge: net work equals the change in kinetic energy, which is what lets you skip the whole force-and-time analysis and jump straight from a distance to a speed. Potential energy is the bookkeeping for work you can get back — lifted mass, compressed spring.

Choose energy over forces whenever the problem gives you a start state and an end state but no clock: a roller coaster, a block sliding down a ramp, an arrow leaving a bow. Choose forces when you need the motion in between. For power, P = W/t is for a job over an interval and P = Fv is for the instantaneous power at a given speed, which is why a car's power output caps its top speed. The most common error is the cos θ in the work formula: a force perpendicular to the motion does no work at all, so carrying a suitcase down a level corridor is, in physics terms, free, however heavy it feels.