Using mathematical terms, terminal speed—without considering buoyancy effects—is given by
where
* represents terminal velocity,
* is the mass of the falling object,
* is the acceleration due to gravity,
* is the drag coefficient,
* is the density of the fluid through which the object is falling, and
* is the projected area of the object.
In reality, an object approaches its terminal speed asymptotically.
Buoyancy effects, due to the upward force on the object by the surrounding fluid, can be taken into account using Archimedes' principle: the mass  has to be reduced by the displaced fluid mass , with  the volume of the object. So instead of  use the reduced mass  in this and subsequent formulas.
The terminal speed of an object changes due to the properties of the fluid, the mass of the object and its projected cross-sectional surface area.
Air density increases with decreasing altitude, at about 1% per  (see barometric formula). For objects falling through the atmosphere, for every  of fall, the terminal speed decreases 1%. After reaching the local terminal velocity, while continuing the fall, speed decreases to change with the local terminal speed.
Using mathematical terms, defining down to be positive, the net force acting on an object falling near the surface of Earth is (according to the drag equation):
with v(t) the velocity of the object as a function of time t.
At equilibrium, the net force is zero (F = 0) and the velocity becomes the terminal velocity :
Solving for V yields:
The drag equation is—assuming ρ, g and C to be constants:
Although this is a Riccati equation that can be solved by reduction to a second-order linear differential equation, it is easier to separate variables.
A more practical form of this equation can be obtained by making the substitution .
Dividing both sides by m gives
The equation can be re-arranged into
Taking the integral of both sides yields
After integration, this becomes
or in a simpler form
with artanh the inverse hyperbolic tangent function.
Alternatively,
with tanh the hyperbolic tangent function. Assuming that g is positive (which it was defined to be), and substituting α back in, the speed v becomes
Using the formula for terminal velocity
the equation can be rewritten as
As time tends to infinity (t → ∞), the hyperbolic tangent tends to 1, resulting in the terminal speed
For very slow motion of the fluid, the inertia forces of the fluid are negligible (assumption of massless fluid) in comparison to other forces. Such flows are called creeping or Stokes flows and the condition to be satisfied for the flows to be creeping flows is the Reynolds number, . The equation of motion for creeping flow (simplified Navier–Stokes equation) is given by:
where:
*  is the fluid velocity vector field,
*  is the fluid pressure field,
*  is the liquid/fluid viscosity.
The analytical solution for the creeping flow around a sphere was first given by Stokes in 1851.
From Stokes' solution, the drag force acting on the sphere of diameter  can be obtained as
where the Reynolds number, . The expression for the drag force given by equation () is called Stokes' law.
When the value of  is substituted in the equation (), we obtain the expression for terminal speed of a spherical object moving under creeping flow conditions:
where  is the density of the object.