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 downhill along a straight track from point 1 to 2 to 3.� As the ball
rolls downhill, it speeds up.� Hence the lengths of the velocity arrows 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 acceleration is in the
same direction as the velocity.�
In summary, the angle between the velocity and acceleration will be 0� whenever an object speeds up while moving in a straight line.�