2
1.1. HOW DO WE MEASURE VELOCITY?
Preview Activity 1.1. Suppose that the height s of a ball (in feet) at time t (in seconds) is
given by the formula s(t) = 64 − 16(t − 1) 2 .
(a) Construct an accurate graph of y = s(t) on the time interval 0 ≤ t ≤ 3. Label at
least six distinct points on the graph, including the three points that correspond
to when the ball was released, when the ball reaches its highest point, and when
the ball lands.
(b) In everyday language, describe the behavior of the ball on the time interval
0 < t < 1 and on time interval 1 < t < 3. What occurs at the instant t = 1?
(c) Consider the expression
AV [0.5,1] =
s(1) − s(0.5)
1 − 0.5
.
Compute the value of AV [0.5,1] . What does this value measure geometrically? What
does this value measure physically? In particular, what are the units on AV [0.5,1] ?
⊲⊳
Position and average velocity
Any moving object has a position that can be considered a function of time. When this
motion is along a straight line, the position is given by a single variable, and we usually
let this position be denoted by s(t), which reflects the fact that position is a function of
time. For example, we might view s(t) as telling the mile marker of a car traveling on a
straight highway at time t in hours; similarly, the function s described in Preview Activity
1.1 is a position function, where position is measured vertically relative to the ground.
Not only does such a moving object have a position associated with its motion, but
on any time interval, the object has an average velocity. Think, for example, about driving
from one location to another: the vehicle travels some number of miles over a certain time
interval (measured in hours), from which we can compute the vehicle’s average velocity. In
this situation, average velocity is the number of miles traveled divided by the time elapsed,
which of course is given in miles per hour. Similarly, the calculation of AV [0.5,1] in Preview
Activity 1.1 found the average velocity of the ball on the time interval [0.5, 1], measured in
feet per second.
In general, we make the following definition: for an object moving in a straight line
whose position at time t is given by the function s(t), the average velocity of the object on the
interval from t = a to t = b, denoted AV [a,b] , is given by the formula
AV [a,b] =
s(b) − s(a)
b − a
.
1.1. HOW DO WE MEASURE VELOCITY?
Preview Activity 1.1. Suppose that the height s of a ball (in feet) at time t (in seconds) is
given by the formula s(t) = 64 − 16(t − 1) 2 .
(a) Construct an accurate graph of y = s(t) on the time interval 0 ≤ t ≤ 3. Label at
least six distinct points on the graph, including the three points that correspond
to when the ball was released, when the ball reaches its highest point, and when
the ball lands.
(b) In everyday language, describe the behavior of the ball on the time interval
0 < t < 1 and on time interval 1 < t < 3. What occurs at the instant t = 1?
(c) Consider the expression
AV [0.5,1] =
s(1) − s(0.5)
1 − 0.5
.
Compute the value of AV [0.5,1] . What does this value measure geometrically? What
does this value measure physically? In particular, what are the units on AV [0.5,1] ?
⊲⊳
Position and average velocity
Any moving object has a position that can be considered a function of time. When this
motion is along a straight line, the position is given by a single variable, and we usually
let this position be denoted by s(t), which reflects the fact that position is a function of
time. For example, we might view s(t) as telling the mile marker of a car traveling on a
straight highway at time t in hours; similarly, the function s described in Preview Activity
1.1 is a position function, where position is measured vertically relative to the ground.
Not only does such a moving object have a position associated with its motion, but
on any time interval, the object has an average velocity. Think, for example, about driving
from one location to another: the vehicle travels some number of miles over a certain time
interval (measured in hours), from which we can compute the vehicle’s average velocity. In
this situation, average velocity is the number of miles traveled divided by the time elapsed,
which of course is given in miles per hour. Similarly, the calculation of AV [0.5,1] in Preview
Activity 1.1 found the average velocity of the ball on the time interval [0.5, 1], measured in
feet per second.
In general, we make the following definition: for an object moving in a straight line
whose position at time t is given by the function s(t), the average velocity of the object on the
interval from t = a to t = b, denoted AV [a,b] , is given by the formula
AV [a,b] =
s(b) − s(a)
b − a
.