Section I. B. [Motivation] Estimating Instantaneous Velocity
Most moving objects (cars, trucks, busses, ships and airplanes, as well as humans) do not
travel at a constant speed. Instead, the speed of these objects varies almost every instant, even
though the changes in speed may not be noticeable. This last statement may seem a little vague,
but consider your experience with automobile speedometers; this should support your
understanding of the concept of instantaneous speed for a moving object.
For example, suppose a car travels 5 miles through city traffic in 20 minutes with a speed that
varies during the trip. Recall from the discussion of average speed in section 0.?? that average
speed is determined by the ratio of change in position to change in time,
. Thus the car's
average speed for the 10-minute time interval is
. The car’s speed at
any particular instant is not likely to be exactly
; sometimes its speed will be faster, and
other times slower, and it may (in fact, will) have that velocity at some instant! Note that, in
contrast, a second car driven on a closed speedway at a constant speed of
for the same
20 minutes will travel the same distance of 5 miles.
Velocity versus speed: As we mentioned previously in section 0.?? , the concept of speed
does not give any information about the direction of travel. It is the velocity of an object, whether
average or instantaneous velocity, which measures both the speed and direction of motion.
Our purpose in this section is to analyze the motion of objects using instantaneous velocity.
In most physical contexts, the average velocity for an
extremely short time interval is a good estimate for the
velocity at any instant during that time period. The
instantaneous velocity at time t =a can be estimated by finding
the average velocity for a time interval between times t =a and t
=x where x is close to, but not equal to, a. That is, we consider
the ratio of the change in position to the length of this short time
interval. We explore this estimation approach further in the next
example.
DPBS on short time intervals: With most
motion that we encounter, the shorter the
time interval, the less the velocity will
vary. How short an interval of time should
we examine so the velocity will be almost
constant? a second? a micro-second ? a
milli-micro second? shorter than that?
Yes!
Example I.B.4. [AThought Experiment] Instantaneous Velocity Derived from
Average Velocity Estimates. The position of an object moving on a coordinate line
at time t seconds is s(t) feet where s(t) = t 2. Estimate the instantaneous velocity of the
object when t = 3 seconds by finding the average velocity for the intervals [2,3],
[2.9,3], [2.99,3], [3,4], [3,3.1], and [3,3.01]. Find a formula for the average velocity
Chapter I.B 4/26/08
1
Figure I.13
of the object for any interval between t = 3 and t = x. Using this formula, discuss what the
instantaneous velocity of this object is at 3 seconds.
Solution [Estimation analysis for velocity]: Let
2
denote the average velocity for the time interval between t x s(x)=x
= 3 and t = x seconds. We initially want to determine
4
-5
-1
,
and finally 2
2.9
8.41
-0.59
-0.1
a formula for
. Table I.14 shows a table with the
relevant information for the computation of these average 2.99 8.9401
-0.0599
-0.01
velocities along with the results. See also Figure
4
16
7
1
I.13.[Add mapping figure for
] Based on this
3.1
9.61
0.61
0.1
information it would seem reasonable to estimate the
instantaneous velocity when t = 3, which we denote by v,
3.01 9.0601
0.0601
0.01
with a number very close to 6, i.e., v. 6.
Next we find
in general (with x =/ 3) using some
Figure I.14
algebra to simplify the
expression:
Comment: Notice the similarity between this example and the tangent
problem estimates in Example I.A.3.
We continue to find the instantaneous velocity for an object with
Figure 2
position S(t)=t 2 meters at time t=a seconds. We denote this
instantaneous velocity by v(a), and the average velocity determined on
the time interval between t = a and t = x by
. We use the same algebra as in Example
I.A.2 to simplify the expression for
, namely
,
Chapter I.B 4/26/08
5.9
5.99
7
6.1
6.01
.
Notice that this formula for the average velocity at time x is consistent
with results we had from the direct computations.
Using the arrow notation for estimation introduced in section I.A.2,
consider what happens to
as x 6 3 with x…3. You can think of
this either statically with x chosen very close to 3 or dynamically, with x
approaching the number 3. With either point of view it should seem
sensible that
. Since the average velocities will be better
estimates of the instantaneous velocity when x is close to 3, we conclude
that the instantaneous velocity at 3 seconds must be 6 meters per second.
Then as x6a, with x…a,
5
= x + a 6 2a.
2
As in the previous discussion where x was 3, when x approaches, or, is close to, a,
is a
6 v(a). Therefore it should make sense that v(a) must
good estimate for v(a), i.e., as x6a,
equal 2a, i.e., v(a) = 2a.
MOVE TO 0.C. Graphs and motion: We use graphs to understand
Add Historical note
motion. To be specific, for an object moving on a straight line we’ll let s(t)
-Oresme and Galileo?
be the coordinate of the object at time t. It is commonly presumed that an
object can be in only one place for a given time, so its position, s(t), is a
function of time. Consider the graph of this position function. See Figure *. The first coordinate
of a point on the graph gives the time value, t, while the second coordinate gives the position
value of the object, s(t). Combined as an ordered pair (t,s(t)) we can describe this point on the
graph as a state of the object. The change in position between the two states, )S, is the change in
the second coordinates of the points, while the change in time for the two states, )t, is the change
in the first coordinates of the points. Thus the average velocity determined by the
two states,
, corresponds to the slope of the non-vertical line
determined by the two points.
On the graph of the position function, the slope of the secant line,
,
determined by the points (a, s(a)) and (x, s(x)) uses the same computations used
to determine the average velocity over the time interval between a and x. Thus we
have
.
[Need transition to connect with the graph of position in ch 0.C for next
paragraph+]
Our analysis of average velocity and instantaneous velocity has suggested that
when x6a, the average velocity determined by x and a approaches the
instantaneous velocity at time a, i.e.,
6 v(a). In Section I.A our work
Figure I.13
suggested that the slope of the secant lines to the graph of the position function, s, determined by
x and a approaches the slope of the line tangent to the curve that graphs the position function,
i.e.,
. Putting all this together gives us the following conclusion:
The numerical value of the instantaneous velocity at time a is precisely the same
as the slope of the line tangent to the graph of the position function at the point (a,
s(a)). Positive velocity is represented by a tangent line with positive slope, going
"uphill," while a negative velocity corresponds to a tangent line with negative
slope, going "downhill." Greater speeds are interpreted visually as steeper tangent
lines.
These relations can be reversed as well. A tangent (or secant) line can be interpreted to give
information about the velocity of a moving object. The following example and discussion shows
more of the connection between instantaneous velocity and the slope of a line tangent to a graph.
Chapter I.B 4/26/08
3
Example I.B.5: Suppose the graph in Figure 4 is the
position function of a cat moving in a narrow chamber
with its distance from one end of the chamber measured in
meters at time t seconds. Based on the graph, estimate the
cats’'s velocity at times 1 and 2 seconds. Estimate the
time(s) at which the cat turned around.
Discussion: To estimate velocity based on the given
graphical information about the cat's position, we consider
the slopes of the lines tangent to the graph as measuring
instantaneous velocities.
Looking at the point on the graph with first coordinate
1 and estimating the slope of a line tangent to the graph at
that point gives an estimate of the instantaneous velocity at
time 1 second. If you draw the tangent line at the point
with first coordinate 1, you will see a line that has a slope
Figure 4
of about -2, so we estimate the cat's velocity as -2 meters
per second. Similarly we estimate the velocity at time 2 by estimating the slope of the tangent
line at the point on the graph with first coordinate 2. This tangent line seems to be horizontal, so
the slope would be 0, meaning a 0 velocity. So the cat appears to be stopped at time 2.
When the cat turned around its position function will change from increasing in value to
decreasing or vice versa. Also his velocity will reverse in sign, and so too will the sign of the
slope of the corresponding tangent line to the graph. Furthermore, for that instant when the cat
turns around it will appear to be standing still. Based on all these criteria we estimate that the cat
turned around approximately when t =.5 since at that point the graph "turns around," the tangent
line slopes change from being positive to being negative, and the slope of the tangent line at this
point appears to be 0.Whereas at time 2 the slope does not change sign so the cat stopped for an
instant then resumed moving in the same direction.
[Add a gravity with cannonball example.]
Chapter I.B 4/26/08
4
Exercises I.B.
1.
2.
Explain the difference between average velocity and instantaneous velocity for a moving
object.
Visualizing estimates for instantaneous velocity. With the power of technology it is
possible to look much more closely at algebraically defined position functions. The
ability to ”zoom in” on data allows more accurate estimates for the instantaneous
velocity.
Suppose the following mapping figures represent the position function S of an
object at time t. For each figure, based on the information given, estimate the
instantaneous velocity at the indicated instant.
[INCLUDE 3 MAPPING FIGURES HERE!]
In problems 3 through 14 assume that s(t) is the position of an object moving on a coordinate
line.
a) Find the average velocity of the object for the time intervals:
[0, 1], [1, 2], [0.9, 1], and [1, 1.1].
Sketch a mapping figure to illustrate the data used for these calculations.
b) Find the instantaneous velocity at times t = 1, t = -1, and t = a.
3.
4.
5.
6.
7.
8.
s(t) = 5t 2 - 4t + 1.
s(t) = 3t 2 - 2t.
s(t) = 7 - t 2.
s(t) = 2 - t - 3t 2.
s(t) = 3t - 2.
s(t) = 3 - 4t.
9.
10.
11.
12.
13.
14.
s(t) = 5.
s(t) = 2t 3 - t .
s(t) = 2t 3 - t 2.
s(t) = t 4.
s(t) = 1/(t + 5) with t … -5.
s(t) = 1/(t - 3) with t … 3.
15. Review your work on problems 7-9. Describe the motion of the object referring to both
the average and instantaneous velocities in your calculations.
16. [General Problem about linear/constant velocity.]
17. A weight attached to a spring is moving up and down. Its position above ground level at
time t seconds is given by s(t) = 2 sin(t) + 3 meters. Estimate the instantaneous velocity of
the weight at time t = 0 using average velocities for the time intervals between 0 and x
where x =.2, .1, .05, -.2, -.1, and -.05 seconds. Discuss briefly what you think the
instantaneous velocity is at time t = 0.
18. Consider the position function s(t) = t 2 as in Example I.B.4.
a.
Complete the following table for finding average velocities related to t = 2.
Chapter I.B 4/26/08
5
x
s(x)=x 2
1.9
1.99
1.999
2.001
2.01
2.1
Table for Exercise 18
b.
c.
Based on this work estimate the instantaneous velocity of the object when t = 2.
Sketch mapping figures that illustrate the data from the table. Use different scales
to make the data easily distinguishable.
19. For each of the following position functions use your calculator to estimate the
instantaneous velocity at time t=0 based on calculating an average velocity for the time
interval [0, 0.001].
a.
s(t) = 2 t.
c.
s(t) = (.5) t
b.
s(t) = 3 t
d.
s(t) = (1/3) t.
The smooth transfer. An important part of a relay race is the passing of the baton between
consecutive runners. For a smooth transfer it is desirable for the runners to have the same
instantaneous velocity at the time of transfer. [There are numerous other contexts where
matching two velocities helps make a transfer go smoothly.]
This context can be analyzed abstractly by considering two objects that are at the same
position and moving at the same instantaneous velocity, at the same time . The following
problems explore this idea a little more specifically.
20. Suppose two objects are moving on a coordinate line. At time t the first object, A, is at
position s(t) = t 2 while the second object, B, is moving at a constant velocity and at time 3
B is at position 5.
a. Find any and all possible velocities for B so that at some time the two objects are at
the same position and traveling at the same velocity. Discuss briefly the strategy you
followed in arriving at your result. [Hint: Let v denote the velocity of B, and note
from example XXX that the velocity of A at time t =a is 2a.]
b. Find any and all lines through the point (3,5) that are tangent to the curve y = x 2.
Sketch a graph to illustrate your result.
c. Discuss the relation between parts a and b.
21. Suppose two objects are moving on a coordinate line. At time t the first object, A, is at
position s(t) = t 2 +12 while the second object, B, is moving at a constant velocity and at
time 2 is at position 10.
Chapter I.B 4/26/08
6
a. Find any and all possible velocities for B so that at some time the two objects are at
the same position and traveling at the same velocity. Discuss briefly the strategy you
followed in arriving at your result.
b. Find any and all lines through the point (2,10) that are tangent to the curve y = x 2 +12.
Sketch a graph to illustrate your result.
c. Discuss the relation between parts a and b.
22. An object moves on a coordinate line with its position at time t seconds determined as s(t)
feet where s(t) = t 2 - 4.
a. Find the instantaneous velocity and speed of the object when t is 3 seconds.
b. Find the instantaneous velocity and speed of the object when t is -3 seconds.
c. Find the instantaneous velocity and speed at those times t when s(t) = 0.
d. Find the time t (if any) when the instantaneous velocity at t will be the same as the
average velocity for the interval [ 1,4 ].
23. An anvil dropped from the top of a mesa has its position s(t) feet above ground level at
time t seconds, where s(t) = -16t 2 + 144 feet.
a. How high above ground level is the anvil after 1 second?
b. What is the (instantaneous) velocity at 1 second?
c. When will the anvil be at ground level?
d. What is the velocity when the anvil reaches ground level?
e. What is the average velocity of the anvil from the release time till the time it reached
ground level?
f. Find a time when the instantaneous velocity of the anvil was equal to its average
velocity found in part e.
24. Stopped for an interval vs. stopped for an instant. Write a short story about something
that is moving and comes to a stop, remains stopped for a period of time and then starts to
move again. In the same story describe a context in which something slows down and
stops for only an instant before starting up again. Discuss the meaning of “stopping” in
terms of average velocity and instantaneous velocity.
25. Historical projects. Two people who were very interested in understanding motion using
a mathematical model were Nicole Oresme (1320-1382) and Galileo Galilei (1564-1642).
Investigate the writings of one of these two scientists and write a short paper based on
your reading, describing the treatment of at least one problem of motion. What approach
was used and how are the results stated?
26. On a trip from San Francisco to Los Angeles on U.S. Route 1, the road conditions for
driving vary dramatically with steep hills and winding curves in some sections followed
Chapter I.B 4/26/08
7
by open straight sections that sometimes have no other entering roads for miles. We left
San Francisco at about 7:45 am and stopped for about 45 minutes at a beach between
Carmel and Big Sur for lunch. Later in the day we turned off the road for about a half
hour later just north of Santa Barbara to look at the ocean and the surf pounding on the
coast. The table in Figure *** shows some of the cities
along the way, the distance travelled from San Francisco
to these cities, and the time we reached these cities on
City/Town
Time
the trip. Draw a mapping figure and a graph that might
Arrived
fit this information. Find the average speed for each of
Santa Cruz
9:40 am
the segments of the trip between these cities. Based on
Monterey
11:25 am
this information, in which segments of the trip do you
think the road was most hazardous? Discuss how you
San Luis Obispo
3:20 pm
dealt with the delays for lunch and sightseeing. Are
Santa Barbara
6:10 pm
there things we may have forgotten to tell you about the
trip that would change your conclusions on where the
Los Angeles
9:25 pm
road was dangerous? Discuss how these might change
your conclusions.
Table 5
27. I forgot to tell you in problem 22 that we took a wrong
turn just south of Santa Barbara that took us 15 minutes to figure out and correct and
took us 8 miles out of the way. How does this effect the average speed you calculated?
28. The odometer and the gas gauge on my car have not worked for years. (I
Time
bought the car used and they didn't work when I bought the car.) This isn't
a problem for me when I drive around town. I just make it a point to fill
9:00 am
the gas tank on Mondays, Thursdays, and Saturdays and I never need
9:45 am
more than 8 gallons. The car has a 13.5 gallon gas tank, so this works
fine. On long trips though I need another technique. What I do is figure I
10:30 am
get about 20 miles a gallon on the highway, so I try to fill my tank every
11:15 am
200 miles to be on the safe side. To estimate how many miles I've driven I
watch my speedometer (which still works fairly accurately) as I drive
12:00 pm
along the highway and try to travel at the same speed for 45 minute
12:45 pm
periods. On long trips I do vary my speed to help break up the monotony.
Here is a table of my speeds that I recorded on a recent trip across the
country. Before I started the table I estimated I had traveled 40 miles
starting with a full tank of gas. Based on this table, when do you think I filled my gas
tank?
29. In many sports, speed is essential to winning competitions or enjoying the activity. Write
a brief essay on one of the following sports and the way that speed is involved and what
speeds are actually involved: waterskiing, snow-boarding, swimming, jogging, diving,
marathon running, football, tennis, soccer, softball, golf, baseball (running bases),
bowling.
Chapter I.B 4/26/08
8
Distance
from S.F.
75 miles
122 miles
230 miles
336 miles
430 miles
Speed
55 mph
60 mph
70 mph
60 mph
65 mph
55 mph
30. Quarter Horse Race.
31. Coming home from work- with mapping figure. Transfer info to table and graph.
32. The Bicycle Ride. The following graph shows the distance of a bicyclist from the starting
point of a 50 mile excursion tour of the Avenue of the Giants from start to finish. Based
on this graph complete the following table and sketch a mapping figure showing some
locations at what you consider the important points in time. Write a short story to go
along with this graph that explains what was happening at some of these times.
explore reversal to accumulate estimates of change based
on velocities,
problems that have other real situations of interest for
estimating distance traveled or time elapsed.
33. Estimate distance traveled based on information
about speed at various times from tables.
34. Estimate position function with TF from Tf of
Figure 5
velocity.
35. Choose graphs of velocity functions based on
position functions.
36. Sketch graph of position function based on graph of velocity.
37. A problem about speed and growth rates-- maybe corn or beans, redwoods, children.
38. Project: Investigate how rockets take off and reach "escape velocity."
39. using lengths to keep track of other variables.
Chapter I.B 4/26/08
9