1. 3.4 Velocity and Other Rates of Change
Today we will study and understand the derivative as a rate of change of a function.

2. What other words can be used for derivative?
Slope of tangent
Instantaneous rate of change
Velocity
Speed
Increase or decrease of a quantity with respect to another quantity

3. Change in position is measured in both
displacement
and
distance
The difference between the starting and ending positions of an object
The total amount of “ground covered” by an object or the total length of its path
If you were to drive 10 miles east
and then 4 miles west
10 miles east
4 miles west
Displacement =
6 miles
Distance =
14 miles

4. And the position graph would look like this:
time
10 miles east
4 miles west
Displacement =
6 miles
Distance =
14 miles

5. Questions for Consideration
What is a position-time graph?
What is a velocity-time graph?
How do features on one graph translate into features on the other?

6. Position-Time Graphs Show an object’s position as a function of time.
x-axis: time
y-axis: position

7. Position-Time Graphs 0 s 1 s 2 s 3 s 4 s 5 s 6 s 7 s 8 s 9 s 10 s
Imagine a ball rolling along a table, illuminated by a strobe light every second.
You can plot the ball’s position as a function of time.
0 s
1 s
2 s
3 s
4 s
5 s
6 s
7 s
8 s
9 s
10 s

8. Position-Time Graphs 1 2 3 4 5 6 7 8 9 10
position (cm)
time (s)

9. Position-Time Graphs What are the characteristics of this graph?
Straight line, upward slope
What kind of motion created this graph?
Constant speed 1 2 3 4 5 6 7 8 9 10
time (s)
position (cm)

10. Position-Time Graphs
Each type of motion has a characteristic shape on a P-T graph.
Constant speed
Zero speed (at rest)
Accelerating (speeding up)
Decelerating (slowing down)
Remember that Speed is the absolute value of your rate of change!

11. Position-Time Graphs
Constant speed is represented by a straight segment on the P-T graph.
time (s)
pos. (m)
Constant speed in positive direction.
time (s)
pos. (m)
Constant speed in negative direction.

12. Position-Time Graphs
Constant speed is represented by a straight segment on the P-T graph.
time (s)
pos. (m)
A horizontal segment means the object is at rest.

13. Position-Time Graphs
Curved segments on the P-T graph mean the object’s speed is changing.
time (s)
pos. (m)
Speeding up in positive direction.
time (s)
pos. (m)
Speeding up in negative direction.

14. Position-Time Graphs
Curved segments on the P-T graph mean the object’s speed is changing.
time (s)
pos. (m)
Traveling in positive direction, but slowing down.
time (s)
pos. (m)
Traveling in negative direction, but slowing down.

15. Position-Time Graphs
The slope of a P-T graph is equal to the object’s velocity in that segment.
slope =
change in y
change in x
time (s)
position (m) 10 20 30 40 50
slope =
(30 m – 10 m)
(30 s – 0 s)
slope =
(20 m)
(30 s)
slope = 0.67 m/s

16. Position-Time Graphs
The following P-T graph corresponds to an object moving back and forth along a straight path. Can you describe its movement based on the graph?
time (s)
position (m) N S

17. X t B A C A … Starts at home (origin) and goes forward slowly
B … Not moving (position remains constant as time progresses)
C … Turns around and goes in the other direction quickly, passing up home

18. During which intervals was he traveling in a positive direction?
During which intervals was he traveling in a negative direction?
During which interval was he resting in a negative location?
During which interval was he resting in a positive location?
During which two intervals did he travel at the same speed?
A) 0 to 2 sec B) 2 to 5 sec C) 5 to 6 sec D) 6 to 7 sec E) 7 to 9 sec F) 9 to 11 sec

19. Graphing w/ Accelerationx
Graphing w/ Acceleration t A B C D
A … Start from rest south of home; increase speed gradually
B … Pass home; gradually slow to a stop (still moving north)
C … Turn around; gradually speed back up again heading south
D … Continue heading south; gradually slow to a stop near the starting point

20. Velocity-Time Graphs
A velocity-time (V-T) graph shows an object’s velocity as a function of time.
A horizontal line = constant velocity.
A straight sloped line = constant acceleration.
Acceleration = change in velocity over time.
Positive slope = positive acceleration.
Not necessarily speeding up!
Negative slope = negative acceleration.
Not necessarily slowing down!

21. Velocity-Time Graphs
A horizontal line on the V-T graph means constant velocity.
time (s)
velocity (m/s) N S
Object is moving at a constant velocity North.

22. Velocity-Time Graphs
A horizontal line on the V-T graph means constant velocity.
time (s)
velocity (m/s) N S
Object is moving at a constant velocity South.

23. Velocity-Time Graphs If an object isn’t moving, its velocity is zero.
time (s)
velocity (m/s) N S
Object is at rest

24. Velocity-Time Graphs
If the V-T line has a positive slope, the object is undergoing acceleration in positive direction.
If v is positive also, object is speeding up.
If v is negative, object is slowing down.

25. Velocity-Time Graphs V-T graph has positive slope. N S
time (s)
velocity (m/s) N S
Positive velocity and positive acceleration: object is speeding up!
time (s)
velocity (m/s) N S
Negative velocity and positive acceleration: object is slowing down.

26. Velocity-Time Graphs
If the V-T line has a negative slope, the object is undergoing acceleration in the negative direction.
If v is positive, the object is slowing down.
If v is negative also, the object is speeding up.

27. Velocity-Time Graphs V-T graph has negative slope. N S
time (s)
velocity (m/s) N S
Positive velocity and negative acceleration: object is slowing down,
time (s)
velocity (m/s) N S
Negative velocity and negative acceleration: object is speeding up! (in negative direction)

28. Average velocity can be found by taking:
Consider a graph of position (distance from a starting point) vs. time.
Average velocity can be found by taking:
time (hours)
position
(miles) B A
The speedometer in your car does not measure average velocity, but instantaneous velocity.
(The velocity at one moment in time.)

29. Velocity is the first derivative of position.
Mr. Murphy’s fall from the 196 foot platform can
be expressed with the equation:
…where t is in seconds and s is measured in feet. Given the above statement, the equation for his velocity is:
…which is expressed in what units?
feet/second

30. Which is easy enough except…32 what?
Find
Which is easy enough except…32 what?
t (seconds)
First, let’s look at the graph of velocity. Note that like the position graph, s(t), the y-axis represents velocity while the x axis represents time
v(t) (feet/second)

31. Enlarging Circles (increase of a quantity with respect to another quantity)

32. Since the slope of a line is based on:
…or in this case:
…we are now talking about:
Is an expression of…
Acceleration
which is a rate of change of velocity.
feet/second2

33. feet/second2
Acceleration is the derivative of velocity.

34. Speed is the absolute value of velocity.
Gravitational
Constants:
Example:
Free Fall Equation
Speed is the absolute value of velocity.

35. Speed is the absolute value of velocity.
Wait! So what is the difference between speed and velocity?
If the object is moving upward…
…or to the right
If the object is moving downward…
…or to the left
But speed does not indicate direction so speed will always be positive…
Speed is the absolute value of velocity.

36. Speed is the absolute value of velocity.
Wait! So what is the difference between speed and velocity?
If an object is moving upward at a speed of 40 ft/sec…
If an object is moving downward at a speed of 40 ft/sec
But regardless of the direction, the speed will always be positive…
Speed is the absolute value of velocity.

37. For motion on a line, what is the relationship between velocity and speed?

38. It is important to understand the relationship between a position graph, velocity and acceleration:
WAIT! How can you tell that acceleration is positive?
The slope (velocity) is tilting upward so the slope (velocity) is increasing
time
position
acc neg
vel pos &
decreasing
acc neg
vel neg &
decreasing
acc zero
vel neg &
constant
acc zero
vel pos &
constant
acc pos
vel neg &
increasing
velocity
zero
acc pos
vel pos &
increasing
acc zero,
velocity zero

39. Based on this graph, how can you use velocity and acceleration to determine when an object is speeding up or slowing down?
time
position
acc neg
vel pos &
decreasing
acc neg
vel neg &
decreasing
acc zero
vel neg &
constant
acc zero
vel pos &
constant
acc pos
vel neg &
increasing
velocity
zero
acc pos
vel pos &
increasing
acc zero,
velocity zero
time

40. Slowing
Down
Speeding up
time
position
acc neg
vel pos &
decreasing
acc neg
vel neg &
decreasing
acc zero
vel neg &
constant
acc zero
vel pos &
constant
acc pos
vel neg &
increasing
velocity
zero
acc pos
vel pos &
increasing
acc zero,
velocity zero
time

41. v and a have the same sign v and a have opposite signs
Slowing
Down
Speeding up
v and a have the same sign
v and a have opposite signs
So the moral of the story is…

42. p Rates of Change: Average rate of change =
Instantaneous rate of change =
These definitions are true for any function.
( x does not have to represent time. ) p

43. Notation for particle motion:

44. The graph shows the position s(t) of a particle moving along a horizontal coordinate axis.
When is the particle moving left?
When is the particle moving to the right?
When is the particle standing still?
When is the particle moving fastest?
Graph the velocity and speed (where defined).
Velocity Speed
Graph: Graph:

45. The graph shows the position s(t) of a particle moving along a horizontal coordinate axis.
When does the particle reverse direction?
When is the particle moving at a constant speed?
When is the particle moving at its greatest speed?
Graph the acceleration (where defined).
Acceleration
Graph:

46. A particle moves along a vertical coordinate axis so that its position at any time t ≥ 0 is given by the function below, where s is measured in centimeters and t is measured in seconds.
Find the displacement during the first 6 seconds.
Find the average velocity during the first 6 seconds.
Find expressions for the velocity and acceleration at time t.
For what values of t is the particle moving downward?

47. Studying Particle Motion – without a calculator
A particle moves along the x-axis so that at time t its position is given by
What is the velocity of the particle at any time t?
During what time intervals is the particle moving to the left?
Justify your answer.
c. At what time on [0, 3] is the particle moving fastest? Justify
your answer.
d. At what time on [0, 4] is the particle moving fastest? Justify

48. Studying Particle Motion – on a calculator…
A particle moves along the x-axis so that at time t its position is given by
What is the velocity
of the particle at any time t?
b. During what time intervals
is the particle moving to the left?
Justify your answer.
c. At what time on [0, 1.4] is
the particle moving fastest?
d. At what time on [0, 2.5] is