Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies.

Velocity is a physical vector quantity: both magnitude and direction are needed to define it.

The scalar absolute value (magnitude) of velocity is called speed, being a coherent derived unit whose quantity is measured in the SI (metric system) as metres per second (m/s or m⋅s−1).

If there is a change in speed, direction or both, then the object is said to be undergoing an acceleration.

The average velocity of an object over a period of time is its change in position,

From this derivative equation, in the one-dimensional case it can be seen that the area under a velocity vs. time (v vs. t graph) is the displacement, s. In calculus terms, the integral of the velocity function v(t) is the displacement function s(t).

Since the derivative of the position with respect to time gives the change in position (in metres) divided by the change in time (in seconds), velocity is measured in metres per second (m/s).

Although velocity is defined as the rate of change of position, it is often common to start with an expression for an object's acceleration.

As seen by the three green tangent lines in the figure, an object's instantaneous acceleration at a point in time is the slope of the line tangent to the curve of a v(t) graph at that point.

In other words, instantaneous acceleration is defined as the derivative of velocity with respect to time:[9]

In the special case of constant acceleration, velocity can be studied using the suvat equations.

By considering a as being equal to some arbitrary constant vector, this shows

It is also possible to derive an expression for the velocity independent of time, known as the Torricelli equation, as follows:

The above equations are valid for both Newtonian mechanics and special relativity.

In particular, in Newtonian 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.

In classical mechanics, Newton's second law defines momentum, p, as a vector that is the product of an object's mass and velocity, given mathematically as

The kinetic energy of a moving object is dependent on its velocity and is given by the equation[10]

Kinetic energy is a scalar quantity as it depends on the square of the velocity.

The general formula for the escape velocity of an object at a distance r from the center of a planet with mass M is[12]

The escape velocity from Earth's surface is about 11 200 m/s, and is irrespective of the direction of the object.

This makes "escape velocity" somewhat of a misnomer, as the more correct term would be "escape speed": any object attaining a velocity of that magnitude, irrespective of atmosphere, will leave the vicinity of the base body as long as it does not intersect with something in its path.

In special relativity, the dimensionless Lorentz factor appears frequently, and is given by[13]

In Newtonian mechanics, the relative velocity is independent of the chosen inertial reference frame.

This is not the case anymore with special relativity in which velocities depend on the choice of reference frame.

In a two-dimensional system, where there is an x-axis and a y-axis, corresponding velocity components are defined as[15]

The magnitude of this vector represents speed and is found by the distance formula as

In three-dimensional systems where there is an additional z-axis, the corresponding velocity component is defined as

While some textbooks use subscript notation to define Cartesian components of velocity, others use

The transverse speed (or magnitude of the transverse velocity) is the magnitude of the cross product of the unit vector in the radial direction and the velocity vector.

If forces are in the radial direction only with an inverse square dependence, as in the case of a gravitational orbit, angular momentum is constant, and transverse speed is inversely proportional to the distance, angular speed is inversely proportional to the distance squared, and the rate at which area is swept out is constant.