Acceleration is defined as a rate of change of the velocity of a moving object:
`veca = (Delta vecv)/(Delta t)`
Note that acceleration, like velocity, is a vector quantity.
When an object moves along a curved trajectory (path), its velocity, in general, changes in both speed and direction. The velocity vector is always tangent to the trajectory. The acceleration vector can point anywhere, and its direction depends on whether the object is speeding up or slowing down.
To describe the acceleration vector when an object is at a given point on a trajectory, it is convenient to break it up into two components: one tangent to trajectory (and co-linear with velocity) at this point, and another one perpendicular to the tangent to trajectory, pointing toward the center of its curvature. The first component is called tangential acceleration, and the second one is centripetal acceleration.
The tangential acceleration indicates the change in the magnitude of the velocity vector, or speed. Thus, if the speed of the object is constant, the tangential acceleration is zero. Otherwise, it equals the rate of change of speed:
`a_t = (Deltav)/(Deltat)`
The tangential acceleration component points in the same direction as velocity if the speed is increasing, and in the opposite direction if the speed is decreasing. For circular motion, it can also be related to the angular acceleration `alpha`
through the formula
`a_t = alpha*R` , where R is the radius of curvature.
The centripetal acceleration indicates the change in the direction of the velocity vector. The greater the curvature of the trajectory, the more extreme the change of the direction of velocity. Thus, the centripetal acceleration is inversely proportional to the radius of the trajectory at the given point. It is also proportional to the square of the speed:
`a_c = v^2/R`
The centripetal acceleration always points toward the center of the curvature of the trajectory. An object moving along along a curved trajectory always has non-zero centripetal acceleration. The only motion with zero centripetal acceleration is motion along a straight line.