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 follows a parabolic trajectory. It speeds up as it goes downward from point 1 to 2 to 3. Hence the lengths of the velocity vectors get progressively longer.

 

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 angle between the acceleration and velocity vectors at point 2 is acute but not 0°.

 

In summary, the angle between the velocity and acceleration will be acute but not 0° whenever an object speeds up while turning. In addition, the acceleration is directed toward the inside of the turn.

 

Having demonstrated the theory above, recall that the acceleration for an object that is flying through a vacuum is the freefall acceleration (i.e., the acceleration of gravity). The freefall acceleration has a magnitude of 9.8m/s² and is directed vertically downward.