Acceleration is the rate of change of velocity. At any point on a trajectory, the magnitude of the acceleration is given by the rate of change of velocity in both magnitude and direction at that point. The true acceleration at time t is found in the limit as time interval Δt → 0. Components of acceleration for a planar curved motion. The tangential component a_t is due to the change in speed of traversal, and points along the curve in the direction of the velocity vector. The centripetal component a_c is due to the change in direction of the velocity vector and is normal to the trajectory, pointing toward the center of curvature of the path.

In physics, and more specifically kinematics, acceleration is the change in velocity over time. Because velocity is a vector, it can change in two ways: a change in magnitude and/or a change in direction. In one dimension, acceleration is the rate at which something speeds up or slows down. However, as a vector quantity, acceleration is also the rate at which direction changes. Acceleration has the dimensions L T⁻². In SI units, acceleration is measured in metres per second squared (m/s²).

In common speech, the term acceleration commonly is used for an increase in speed (the magnitude of velocity); a decrease in speed is called deceleration. In physics, a change in the direction of velocity also is an acceleration: for motion on a planar surface, the change in direction of velocity results in centripetal acceleration; whereas the rate of change of speed is a tangential acceleration.

In classical mechanics, the acceleration of a body is proportional to the resultant (total) force acting on it (Newton's second law):

$\backslash mathbf\{F\}\; =\; m\backslash mathbf\{a\}\; \backslash quad\; \backslash to\; \backslash quad\; \backslash mathbf\{a\}\; =\; \backslash mathbf\{F\}/m$

where F is the resultant force acting on the body, m is the mass of the body, and a is its acceleration.

Tangential and centripetal acceleration

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

$\backslash boldsymbol\; v\; (t)\; =v(t)\; \backslash frac\; \{\; \backslash boldsymbol\; v\; (t)\}\{v(t)\}\; =\; v(t)\; \backslash mathbf\{u\_t\}(t)\; ,$

with v(t) equal to the speed of travel along the path, and

$\backslash mathbf\{u\_t\}\; =\; \backslash frac\; \{\backslash boldsymbol\; v(\; t)\}\{v(t)\}\; \backslash \; ,$

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 u_t, the acceleration of a particle moving on a curved path on a planar surface can be written using the chain rule of differentiation as:

$\backslash begin\{alignat\}\{3\}$

\mathbf{a} & = \frac{d \boldsymbol v}{dt} \\