6
1.1. HOW DO WE MEASURE VELOCITY?
Now, returning to our computation of the average velocity, we find that
AV [0.5,0.5+h] =
s(0.5 + h) − s(0.5)
(0.5 + h) − 0.5
=
(12 − 16h − 16h 2 ) − (16 − 16(0.5) 2 )
0.5 + h − 0.5
=
12 − 16h − 16h 2 − 12
h
=
−16h − 16h 2
h
.
At this point, we note two things: first, the expression for average velocity clearly depends
on h, which it must, since as h changes the average velocity will change. Further, we note
that since h can never equal zero, we may further simplify the most recent expression.
Removing the common factor of h from the numerator and denominator, it follows that
AV [0.5,0.5+h] = −16 − 16h.
Now, for any small positive or negative value of h, we can compute the average velocity.
For instance, to obtain the average velocity on [0.5, 0.75], we let h = 0.25, and the average
velocity is −16 − 16(0.25) = −20 ft/sec. To get the average velocity on [0.4, 0.5], we let
h = −0.1, which tells us the average velocity is −16 − 16(−0.1) = −14.4 ft/sec. Moreover,
we can even explore what happens to AV [0.5,0.5+h] as h gets closer and closer to zero. As h
approaches zero, −16h will also approach zero, and thus it appears that the instantaneous
velocity of the ball at t = 0.5 should be −16 ft/sec.
Activity 1.3.
For the function given by s(t) = 64 − 16(t − 1) 2 from Preview Activity 1.1, find the
most simplified expression you can for the average velocity of the ball on the interval
[2, 2 + h]. Use your result to compute the average velocity on [1.5, 2] and to estimate
the instantaneous velocity at t = 2. Finally, compare your earlier work in Activity 1.1.
⊳
Summary
In this section, we encountered the following important ideas:
• The average velocity on [a, b] can be viewed geometrically as the slope of the line
between the points (a, s(a)) and (b, s(b)) on the graph of y = s(t), as shown in Figure 1.2.
• Given a moving object whose position at time t is given by a function s, the average
1.1. HOW DO WE MEASURE VELOCITY?
Now, returning to our computation of the average velocity, we find that
AV [0.5,0.5+h] =
s(0.5 + h) − s(0.5)
(0.5 + h) − 0.5
=
(12 − 16h − 16h 2 ) − (16 − 16(0.5) 2 )
0.5 + h − 0.5
=
12 − 16h − 16h 2 − 12
h
=
−16h − 16h 2
h
.
At this point, we note two things: first, the expression for average velocity clearly depends
on h, which it must, since as h changes the average velocity will change. Further, we note
that since h can never equal zero, we may further simplify the most recent expression.
Removing the common factor of h from the numerator and denominator, it follows that
AV [0.5,0.5+h] = −16 − 16h.
Now, for any small positive or negative value of h, we can compute the average velocity.
For instance, to obtain the average velocity on [0.5, 0.75], we let h = 0.25, and the average
velocity is −16 − 16(0.25) = −20 ft/sec. To get the average velocity on [0.4, 0.5], we let
h = −0.1, which tells us the average velocity is −16 − 16(−0.1) = −14.4 ft/sec. Moreover,
we can even explore what happens to AV [0.5,0.5+h] as h gets closer and closer to zero. As h
approaches zero, −16h will also approach zero, and thus it appears that the instantaneous
velocity of the ball at t = 0.5 should be −16 ft/sec.
Activity 1.3.
For the function given by s(t) = 64 − 16(t − 1) 2 from Preview Activity 1.1, find the
most simplified expression you can for the average velocity of the ball on the interval
[2, 2 + h]. Use your result to compute the average velocity on [1.5, 2] and to estimate
the instantaneous velocity at t = 2. Finally, compare your earlier work in Activity 1.1.
⊳
Summary
In this section, we encountered the following important ideas:
• The average velocity on [a, b] can be viewed geometrically as the slope of the line
between the points (a, s(a)) and (b, s(b)) on the graph of y = s(t), as shown in Figure 1.2.
• Given a moving object whose position at time t is given by a function s, the average