The definition of average acceleration is:
According to this definition, the vector average acceleration
is obtained by dividing a vector by a scalar. Since the scalar (
)
is always positive, this definition indicates that the two vectors (
and
)
have the same direction. We would like to apply this definition to find the
direction of the acceleration at point D. Then and
in
the definition correspond to an interval that spans point D, i.e., an interval with an initial point just before
point D and with a final point just after point D. The diagrams below illustrate this idea
graphically.
Suppose
that a ball rolls uphill from point 1 to 2, then downhill from point 2 to 3.
Since point 2 is at the top of the hill, the speed at point 2 is the slowest.
Hence the length of 
is
the shortest. Assume that point 2 is at the exact center of the interval from
point 1 to 3. Then the speeds at point 1 and 3 are equal.
Recall that the change in velocity between the final point (point 3) and the initial point (point 1) is:
.
This is equivalent to saying that is
the quantity that must be added to the initial velocity in order to obtain the
final velocity:
The diagram
at right shows the vector (the red vector) that must be graphically added to vector
to
obtain 
.
Hence, the red vector must be .
According to the definition of average acceleration,
Since is
a positive scalar, the direction of the acceleration must be the same as the
direction of the change in velocity. The relative magnitudes of and
depend
on the value of .
If the
interval is short enough, then the average acceleration for the interval is
equal to the instantaneous acceleration at the midpoint of the interval, point
2. This is illustrated in the diagram at right. Notice that the resulting
acceleration vector is perpendicular to the velocity at point 2.
In summary, the acceleration will be perpendicular to the velocity whenever an object turns but is not speeding up or slowing down. In addition, the acceleration is directed toward the interior of the turn.