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.

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