In mechanics, acceleration is the rate of change of the velocity of an object with respect to time.

If the vehicle turns, an acceleration occurs toward the new direction and changes its motion vector.

The acceleration of the vehicle in its current direction of motion is called a linear (or tangential during circular motions) acceleration, the reaction to which the passengers on board experience as a force pushing them back into their seats.

When changing direction, the effecting acceleration is called radial (or centripetal during circular motions) acceleration, the reaction to which the passengers experience as a centrifugal force.

If the speed of the vehicle decreases, this is an acceleration in the opposite direction of the velocity vector (mathematically a negative, if the movement is unidimensional and the velocity is positive), sometimes called deceleration[4][5] or retardation, and passengers experience the reaction to deceleration as an inertial force pushing them forward.

Each of these accelerations (tangential, radial, deceleration) is felt by passengers until their relative (differential) velocity are neutralised in reference to the acceleration due to change in speed.

An object's average acceleration over a period of time is its change in velocity,

In the terms of calculus, instantaneous acceleration is the derivative of the velocity vector with respect to time:

(Here and elsewhere, if motion is in a straight line, vector quantities can be substituted by scalars in the equations.)

By the fundamental theorem of calculus, it can be seen that the integral of the acceleration function a(t) is the velocity function v(t); that is, the area under the curve of an acceleration vs. time (a vs. t) graph corresponds to the change of velocity.

An object moving in a circular motion—such as a satellite orbiting the Earth—is accelerating due to the change of direction of motion, although its speed may be constant.

In classical mechanics, for a body with constant mass, the (vector) acceleration of the body's center of mass is proportional to the net force vector (i.e. sum of all forces) acting on it (Newton's second law):

The velocity of a particle moving on a curved path as a function of time can be written as:

a unit vector tangent to the path pointing in the direction of motion at the chosen moment in time.

Taking into account both the changing speed v(t) and the changing direction of ut, the acceleration of a particle moving on a curved path can be written using the chain rule of differentiation[7] for the product of two functions of time as:

where un is the unit (inward) normal vector to the particle's trajectory (also called the principal normal), and r is its instantaneous radius of curvature based upon the osculating circle at time t. The components are called the tangential acceleration and the normal or radial acceleration (or centripetal acceleration in circular motion, see also circular motion and centripetal force), respectively.

Geometrical analysis of three-dimensional space curves, which explains tangent, (principal) normal and binormal, is described by the Frenet–Serret formulas.

Because of the simple analytic properties of the case of constant acceleration, there are simple formulas relating the displacement, initial and time-dependent velocities, and acceleration to the time elapsed:[10]

where In particular, the motion can be resolved into two orthogonal parts, one of constant velocity and the other according to the above equations.

As Galileo showed, the net result is parabolic motion, which describes, e.g., the trajectory of a projectile in vacuum near the surface of Earth.

[11] In uniform circular motion, that is moving with constant speed along a circular path, a particle experiences an acceleration resulting from the change of the direction of the velocity vector, while its magnitude remains constant.

The derivative of the location of a point on a curve with respect to time, i.e. its velocity, turns out to be always exactly tangential to the curve, respectively orthogonal to the radius in this point.

is a vector from the centre of the circle to the particle with magnitude equal to this distance, and considering the orientation of the acceleration towards the center, yields

of a particle may be expressed as an angular speed with respect to a point at the distance

This acceleration and the mass of the particle determine the necessary centripetal force, directed toward the centre of the circle, as the net force acting on this particle to keep it in this uniform circular motion.

In a nonuniform circular motion, i.e., the speed along the curved path is changing, the acceleration has a non-zero component tangential to the curve, and is not confined to the principal normal, which directs to the center of the osculating circle, that determines the radius

In a two-dimensional system, where there is an x-axis and a y-axis, corresponding acceleration components are defined as[12]

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

Newtonian mechanics is exactly revealed to be an approximation to reality, valid to great accuracy at lower speeds.

Unless the state of motion of an object is known, it is impossible to distinguish whether an observed force is due to gravity or to acceleration—gravity and inertial acceleration have identical effects.