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Velocity is a vector measurement of the rate and direction of motion. The scalar absolute value (magnitude) of velocity is speed. Velocity can also be defined as rate of change of displacement or just as the rate of displacement, depending on how the term displacement is used (see article on displacement).

In mechanics the average speed v of an object moving a distance d during a time interval t is described by the simple formula:

${\displaystyle v=d/t\,}$.

The instantaneous velocity vector v of an object whose position at time t is given by x(t) can be computed as the derivative

${\displaystyle \mathbf {v} =d\mathbf {x} /dt\,}$.

Acceleration is the change of an object's velocity over time. The average acceleration of a of an object whose speed changes from v_i to v_f during a time interval t is given by:

${\displaystyle a=(v_{f}-v_{i})/t\,}$.

The instantaneous acceleration vector a of an object whose position at time t is given by x(t) is

${\displaystyle \mathbf {a} =d^{2}\mathbf {x} /(dt)^{2}\,}$

The final velocity v_f of an object which starts with velocity v_i and then accelerates at constant acceleration a for a period of time t is:

${\displaystyle v_{f}=v_{i}+at\,}$

The average velocity of an object undergoing constant acceleration is (v_f + v_i)/2. To find the displacement d of such an accelerating object during a time interval t, substitute this expression into the first formula to get:

${\displaystyle d=t(v_{f}+v_{i})/2\,}$

When only the object's initial velocity is known, the expression

${\displaystyle d=v_{i}t+(at^{2})/2\,}$

can be used. These basic equations for final velocity and displacement can be combined to form an equation that is independent of time:

${\displaystyle v_{f}^{2}=v_{i}^{2}+2ad\,}$

The above equations are valid for both classical mechanics and special relativity. Where classical mechanics and special relativity differ is in how different observers would describe the same situation. In particular, in classical mechanics, all observers agree on the value of t and the transformation rules for position create a situation in which all non-accelerating observers would describe the acceleration of an object with the same values. Neither is true for special relativity.

The kinetic energy (movement energy) of a moving object is linear with both its mass and the square of its velocity:

${\displaystyle E_{v}={\begin{matrix}{\frac {1}{2}}\end{matrix}}mv^{2}}$

The kinetic energy is a scalar quantity.