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.�