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Velocity Operational definition:
dt
v =
Three flavors of velocityuniformaverageinstantaneous
Uniform motion
d = 60 cm
t = 15 sec
d = 60 cm
t = 15 sec
Interpret: 60/15 = 4 cm/sec
d = 60 cm d = 60 cm
t = 15 sec t = 15 sec
Velocity Operational definition:
dt
v =
Three flavors of velocityuniformaverageinstantaneous
Graphing uniform motion
d = 60 cm d = 60 cm
t = 15 sec t = 15 sec
A graph is a collection of points
Time (sec)
Position (cm)
5 10 15 20 25 30
60
120
00
Uniform motion
A graph is a collection of points
Time (sec)5 10 15 20 25 30
60
120
00
Position (cm)
Uniform motion
d = 60 cm
t = 15 sec
A graph is a collection of points
Time (sec)5 10 15 20 25 30
60
120
00
Position (cm)
A graph is a collection of points
Time (sec)5 10 15 20 25 30
60
120
00
Position (cm)
Uniform motion
d = 60 cm d = 60 cm
t = 15 sec t = 15 sec
A graph is a collection of points
Time (sec)5 10 15 20 25 30
60
120
00
Position (cm)
A graph is a collection of points
Time (sec)5 10 15 20 25 30
60
120
00
Position (cm)
A graph is a collection of points
Time (sec)5 10 15 20 25 30
60
120
00
Position (cm)
How do we find the velocity?
Time (sec)5 10 15 20 25 30
60
120
00
rise d
run t
slope = riserun = d
tvelocity =
= 4 cm/s
Position (cm)
40
100
= 60 cm
= 15 s
Velocity Operational definition:
dt
v =
Three flavors of velocityuniformaverageinstantaneous
Average velocity Left home at 5:00pm
At 6:00pm I was 60 miles from home
What was my velocity at 5:30pm?
Operational definition of average velocity
dt
vave =
Interpretation of average velocity
Start Finish
The hare’s average velocity is the uniform velocity of the tortoise. In other words, how fast would the tortoise have to plod along to start and finish the race at the same time as the hare?
Interpretation of average velocity
Start Finish
Problem: If the hare traveled at a uniform 10 miles/hour for the first hour and a uniform 4 miles/hour for the last two hours, what was the hare’s average velocity?
What was the tortoise’s uniform velocity?
Does this bullet have a velocity at the instant shown?
picture courtesy of the late, great Doc Edgerton
Velocity Operational definition:
dt
v =
Three flavors of velocityuniformaverageinstantaneous
What is the velocity of this truck?
t = 4 sec t = 4 sec
Time (sec)
Posit
ion
(m
)
0
5
10
15
20
25
0 1 2 3 4 5
What is the velocity at t = 2 sec?
rise d
run t
slope = riserun = d
tinstantaneous velocity =
Pos
itio
n (m
)
Time (sec)
Posit
ion
(m
)
0
5
10
15
20
25
0 1 2 3 4 5
Why use the tangent line?
Pos
itio
n (m
)
Let’s take a closer look
Time (sec)
Posit
ion
(m
)
0
5
10
15
20
25
0 1 2 3 4 5
Pos
itio
n (m
)
Let’s take a closer look
Time (sec)
Posit
ion
(m
)
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
1.5
1.6
1.7
1.8
1.9 2
2.1
2.2
2.3
2.4
2.5
Pos
itio
n (m
)
Perhaps even closer
Time (sec)
Posit
ion
(m
)
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
1.5
1.6
1.7
1.8
1.9 2
2.1
2.2
2.3
2.4
2.5
Pos
itio
n (m
)
This is close enough
Time (sec)
Po
sit
ion
(m
)
3.5
3.6
3.7
3.8
3.9
4
4.1
4.2
4.3
4.4
4.5
1.9
1.9
2
1.9
4
1.9
6
1.9
8 2
2.0
2
2.0
4
2.0
6
2.0
8
2.1
The slope of this line is 4 m/s
What interpretation can we give to this slope?
Pos
itio
n (m
)
Time (sec)
Po
sit
ion
(m
)
3.5
3.6
3.7
3.8
3.9
4
4.1
4.2
4.3
4.4
4.5
1.9
1.9
2
1.9
4
1.9
6
1.9
8 2
2.0
2
2.0
4
2.0
6
2.0
8
2.1
Now extend this line in both directions and it becomes the tangent
Pos
itio
n (m
)
Now extend this line in both directions and it becomes the tangent
Time (sec)
Po
sit
ion
(m
)
3.5
3.6
3.7
3.8
3.9
4
4.1
4.2
4.3
4.4
4.5
1.9
1.9
2
1.9
4
1.9
6
1.9
8 2
2.0
2
2.0
4
2.0
6
2.0
8
2.1
Pos
itio
n (m
)
Time (sec)
Posit
ion
(m
)
0
5
10
15
20
25
0 1 2 3 4 5
Instantaneous velocity
Pos
itio
n (m
)
Consider the following statement: “d/t gives the velocity for an
interval. If we want the velocity at just one instant, we would divide the position at that instant by the time at that instant: v = d/t.” Is this right?
It’s 12 noon and I’m in Silver Bay.....how fast am I going?
How do you move a pencil to create the following graph?
time
pos
itio
n
What is the corresponding velocity versus time graph?
time
velo
city
How do you move a pencil to create the following graph?
time
pos
itio
n
What is the corresponding velocity versus time graph?
time
velo
city
How do you move a pencil to create the following graph?
time
pos
itio
n
What is the corresponding velocity versus time graph?
time
velo
city
Slope = ?
time
pos
itio
n
Slope = ?
time
pos
itio
n
0
5
10
15
20
25
30
0 1 2 3 4 5
Time (sec)
Posi
tion (m
)
What’s the velocity at t=0?
Graphing velocity
Time (sec)
Vel
ocit
y (m
/s)
1 2 3 4 5 6
2
4
00
6
8
0
5
10
15
20
25
30
0 1 2 3 4 5
Time (sec)
Posi
tion (m
)
What’s the velocity at t=1?
Graphing velocity
Time (sec)1 2 3 4 5 6
2
4
00
6
8
Vel
ocit
y (m
/s)
0
5
10
15
20
25
30
0 1 2 3 4 5
Time (sec)
Posi
tion (m
)
What’s the velocity at t=3?
Graphing velocity
Time (sec)1 2 3 4 5 6
2
4
00
6
8
Vel
ocit
y (m
/s)
0
5
10
15
20
25
30
0 1 2 3 4 5
Time (sec)
Posi
tion (m
)
What’s the velocity at t=4?
Graphing velocity
Time (sec)1 2 3 4 5 6
2
4
00
6
8
Vel
ocit
y (m
/s)
Graphing velocity
Time (sec)1 2 3 4 5 6
2
4
00
6
8
Vel
ocit
y (m
/s)
How do you move a pencil to create the following graph?
time
velo
city
How do you move a pencil to create the following graph?
time
velo
city
+5
-5
0
Velocity
Read Chapter 2
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