The
average (denoted by < >) acceleration, is calculated from a finite
change in velocity vector (as in the sketched example) divided by the time
required to make the change. In the limit of small time interval, the
instantaneous acceleration vector is the time derivative of the velocity
vector.If the velocity and acceleration always lie parallel to the same direction, the motion is described as "one dimensional." In this case we can draw a horizontal line to represent the directions of the vectors. In that case, only two directions (to the right or to the left in our example) are allowed. These directions can be specified as the + and the - directions. Traditionally + is take to be to the right, as in our example.
In the one dimensional case, ordinary algebra is sufficient to keep track of the directions, with + and - taking on their ordinary algebraic meanings.
If the acceleration does not lie along the velocity direction, the problem must be treated using vector algebra, with vector subtraction defined as in the sketch, similar to vector subtraction for position vectors. For calculational puposes it is useful to reduce the problem to a set of one-dimensional problems, using components.
One special case of interest is "opposite" to the case in which acceleration and velocity vectors lie along the same line: In this case, the acceleration is always perpendicular to the velocity. This means that the magnitude of the velocity always stays the same. The effect of the acceleration is exclusively to change the direction of the velocity.
If, in addition, the magnitude of the the acceleration is constant, then the motion has a circular path.