Work, Energy, and Power

Translational kinetic energy is the energy an object has because it is moving, K = ½mv², measured in joules. It's a scalar that's never negative, it quadruples when the speed doubles, and observers in different reference frames can measure different values for it.

Work is energy that a force moves into or out of a system as it acts over a distance: W = F∥d = Fd cos θ. Only the part of the force along the displacement does work, so work can be positive, negative or zero. The work–energy theorem says the net work on an object equals its change in kinetic energy, and on a force–position graph the work is the area under the curve.

Potential energy is energy stored in the arrangement of a system whose objects push or pull on each other with conservative forces, like gravity or a spring. Near Earth's surface ΔU_g = mgΔy, and a spring stretched or squeezed by x stores U_s = ½kx². Only changes in potential energy matter, so near Earth you can pick where U_g = 0; for planets, moons or stars U_g = −Gm₁m₂/r, which puts zero at infinite separation.

Mechanical energy is a system's kinetic plus potential energy, and it stays constant when no outside work is done on the system and no friction-like forces act inside it. Energy is always conserved overall: any change in a system's total energy equals the energy transferred in or out by work, and friction or air resistance turns mechanical energy into thermal energy or sound.

Power is how fast energy is transferred or converted: average power is P = ΔE/Δt = W/Δt, measured in watts (1 W = 1 J/s). The power a force delivers to a moving object at a given instant is the force component along the velocity times the speed, P = F∥v.