The position of a ball thrown into the air is observed to be described by the equation
,
where y is in feet. We wish to determine the equation of its velocity, , and its
acceleration, . To find the velocity, we can differentiate using the Power, Scalar Multiple, and Sum rules to get

From this we see that at t = 0, the ball is moving upward at 12 ft/sec. In contrast, at t = 2 sec, the ball has a velocity of 12 - 32(2) = -52 ft/sec. The negative value means that the ball is travelling downward, after having reached its maximum height.

To find the acceleration, we differentiate again:

The negative sign means that the direction of the acceleration is downward.

Note that this is a special case. When the highest power in y(t) is the 2^nd power, the acceleration will always be a constant value.

Test Problem: Determine the equations for the velocity and acceleration for
.

Select the velocity from
a) ft/sec
b) ft/sec
c) ft/sec
d) ft/sec

Select the acceleration from
e) 40
f) -20
g) 9
h) -40