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 which is a turning
point.� 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 along a straight track from point 1 to 2.� Point 2 is
a turning point.� Afterward, the ball rolls back downhill from point 2 to 3.�
Points 1 and 3 are the same point.� As the ball rolls uphill, it slows down.�
As it rolls downhill, it speeds up.� .�
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 the �and �depends 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 acceleration
points down the incline.�
In summary, the acceleration at a turning point is not zero.� It points down the incline.