Acceleration

In physics, acceleration is the second derivative of the position of a point in space with respect to time. Conventionally acceleration (a vector) is designated by a, the position of the point by r, and time by t, then

$\mathbf {a} \,{\stackrel {\mathrm {def} }{=}}\,{\frac {d^{2}\mathbf {r} }{dt^{2}}}.$

Alternatively, the velocity of the point may be introduced,

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

and the acceleration can be defined as an increase (per unit time) in the velocity,

$\mathbf {a} ={\frac {d\mathbf {v} }{dt}}.$

If we are not considering a point, but a body of finite extent, then 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 definition just given then applies to the position r of the center of mass and the translational motion 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 a second derivative with respect to time.

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 independent of the force, depending only on the body; 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.