Acceleration

In common parlance, the acceleration of an object is the increase of its speed per unit time. In daily language the term acceleration is only used for an increase in speed; a decrease in speed is usually called deceleration.

In physics, speed is the absolute value (magnitude) of velocity, a vector. Physicists define velocity of a point in space as the derivative of the position-vector of the point with respect to time. Conventionally, the position of a point is designated by by r (a vector), velocity by v, acceleration (a vector) by a, and time by t (a scalar). Hence

$\mathbf {v} \,{\stackrel {\mathrm {def} }{=}}\,{\frac {d\mathbf {r} }{dt}}.$

The acceleration a is the derivative of v with respect to time,

$\mathbf {a} \,{\stackrel {\mathrm {def} }{=}}\,{\frac {d\mathbf {v} }{dt}}.$

Accordingly, acceleration is the second derivative of the position of a point in space with respect to time,

$\mathbf {a} ={\frac {d^{2}\mathbf {r} }{dt^{2}}}.$

The direction and length of a may vary from one instant to the other. Since it is not meaningful to compare two non-parallel vectors, the term deceleration is hardly ever used in physics. The second derivative of r, whatever its magnitude or direction, is referred to as acceleration. The unit of acceleration is length per time squared (in SI: m s⁻²).

If the object is not a point, but a body of finite extent, we recall that the motion of the body can be separated in a translation of the center of mass and a rotation around the center of mass. The definitions just given then apply to the position r of the center of mass and the translational velocity and translational acceleration of the center of mass of the body.

The rotational motion of the body is somewhat more difficult to describe. In particular one can prove that there cannot exist in three dimensions a set of angular coordinates such that their first derivative with respect to time is the angular velocity, see rigid rotor for more details. Accordingly, angular acceleration cannot be given as second derivatives of some coordinates with respect to time (except for the special case of rotation around an axis fixed in space).

One of the fundamental laws of physics is Newton's second law. This states that the acceleration of the center of mass of a (rigid) body is proportional to the force acting on the body. The relation between force and acceleration being linear, the proportionality constant is a property of the body only; it is the mass of the body.

In classical mechanics it is frequently the case that the force on a body is proportional to the gradient ∇ of a potential V,

$\mathbf {F} =-m{\boldsymbol {\nabla }}{\big (}V(\mathbf {r} ){\big )},$

with m the total mass of the body. Comparing with Newton's second law, we see that −∇ V is the acceleration of the body (provided m∇V is the only force operative on the body). An example of an acceleration due to a potential is the acceleration due to gravity.