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 goes upward from point 1 to 2,
then downward from point 2 to 3.� Point 2 is at the apex of the parabola so 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.�
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.�