The following definition of acceleration will be for a mass point. See Comment 3 at the end about what it means to talk about the "acceleration of an extended object".

Definition of acceleration in words:

Acceleration is the time-rate-of-change of velocity.

Definition of acceleration in symbols:

In this definition, denotes the acceleration at time t and the velocity vector of the mass point. The symbol Δ denotes the change in velocity between the instant at time t and a later instant at time t + Δt,

Δ = (t + Δt) - (t).

The symbol d/dt is calculus notation for the more explicit notation with the limit symbol. When the notation d/dt is used in physics, it is useful to think of d and dt as very small quantities and of the symbol d/dt as the ratio of these small quantities.

Acceleration is a vector quantity, arising from ratios that have a vector in the numerator and a scalar in the denominator. The direction of the acceleration vector at a given instant is the direction in which the tip of the velocity vector is moving at that instant if the tail end of the velocity vector is being held fixed. (Otherwise, the motion of the tip of the velocity vector would not represent changes Δ in the velocity vector, as are needed in the definition of acceleration.)

With the tail end of the velocity vector held fixed, the tip of the vector describes a path. Let's call it the velocity path. At any given moment, the acceleration vector's direction is along the tangent to the velocity path at that instant. See the following illustration which shows the velocity path of a mass point in magenta, three velocity vectors (in magenta) corresponding to three different instants of time, and the acceleration vectors (in orange) at these three instants.

The magnitude of the acceleration is a scalar quantity denoted either by a or ||. It has no special name.

The SI-unit of acceleration is m/s².

Comment 1. The ratio Δ/Dt behind the limit operation in the definition of acceleration is called average acceleration. With this concept, the definition of acceleration can be reworded as follows.

The acceleration at time t is the limit of the average acceleration over a time interval extending from time t to time t + Δt when Dt goes to zero.

Comment 2. When a motion is along a straight line, say, the x-axis, most of the time one will be working with the x-component a_x of the acceleration instead of the vector and will be referring to a_x as "acceleration" instead of "x-component of acceleration", for brevity's sake. One may even drop the subscript x and write just a instead of a_x, again for brevity's sake. However, one should explain that a denotes the (x-component of the) acceleration, not the magnitude of the acceleration, unless this is clear from the context.

Comment 3. When an extended object is moving, its parts may be moving with different accelerations, e.g., the object may be rotating or the object may be vibrating. In such cases, one cannot just speak of "the acceleration of the object".

In many cases, however, it will be possible to assume the object to be rigid and non-rotating. In such cases, all parts of the object have the same acceleration and it makes sense to speak of the acceleration of the object. In other cases, e.g., when the object is rotating in addition to performing some motion as a whole, one will mean the acceleration of the center of mass of the object when speaking of the object's acceleration. This should be made clear, however, unless it is clear from the context.

Comment 4. When a particle is moving in a circle of radius r with constant speed v (uniform circular motion), the direction of the acceleration is radial, from the particle towards the center of the particle's circular orbit, and the magnitude of the acceleration is equal to

a = v² / r.

The acceleration vector, as always, is tangential to the velocity path, but it is radial to the position path in this case.