Friction does not conserve energy, since momentum is lost to the medium in the form of heat. It is irreversible: energy is lost by the object but not regained. Friction forces must change signs under , hence they must depend on velocity; in doubt, a friction force should slow the object down.

As far as we know, the fundamental laws of nature are invariant under CPT, so friction forces cannot be fundamental forces.

Linear drag, , models viscous effects where objects move the medium with them, such as a spoon in honey. For a spherical object of radius , Stokes’ law gives , where is the viscosity of the medium.

Quadratic drag, , arises when an object bumps into molecules: the collision rate is proportional to the speed , and each collision imparts a momentum change proportional to , so the force is proportional to . The number of collisions depends on the density of the medium and the cross-sectional area of the object, so . By dimensions,

Both linear and quadratic drag are typically present; and are called coefficients of friction.

For a spherical object, the ratio of quadratic to linear drag is

the Reynolds number, which decides which term dominates.

For a particle falling with quadratic friction under gravity, the -component of the motion obeys

The velocity starts at and increases; initially the right-hand side is dominated by , but eventually the two forces balance, giving the terminal velocity

so heavier objects have higher terminal velocities. By dimensional analysis, the timescale to reach it is

Friction damps small oscillations about an equilibrium point, where linear drag dominates. In a 1D system,

where is the natural frequency of oscillations without friction and is the damping coefficient. Solving,

and taking the real part gives the damped oscillations. The three cases are:

• : underdamped, decaying oscillations.

• : overdamped, exponential decay.

• : critical damping, .