1. Motion Graphs

2. Motion Graphs
Pictorial representations of data (or predictions/calculations)
There are two motion equations that are normally graphed
Position Equation
Posf = Poso + Vot +(1/2)at2 (General equation)
Sf = So + Vot (a = 0 so Vo = Vany = V)
Velocity Equation
Vf = Vo + at (General equation)
Vf = Vo (a = 0 this is a trivial case)

3. Role of slope All slopes are a ratio of changes (m =DY/DX)
In other words a slope represents the rate that the Y axis changes with respect to how the X axis changes.
For many motion graphs the X axis represents time
Position vs.. Time graph (slope = velocity)
Velocity vs.. Time graph (slope = acceleration)
There are often 4 cases that have to be recognized
Increasing slope
Decreasing slope
Constant slope
No slope

4. Recognizing the types of slopes
Increasing Slope
(Moves away from 0)
Graphs of slopes 2 4 6 1 3 5 2 4 6 1 3 5

5. Recognizing types of slope P2
Decreasing Slope
(Moves toward 0)
Graphs of slopes 2 4 6 1 3 5 2 4 6 1 3 5

6. Recognizing Types of Slopes P3
Constant Slope
Graphs of slopes 2 4 6 1 3 5 2 4 6 1 3 5

7. Recognizing Types of Slopes P4
No Slope
Graphs of slopes 2 4 6 1 3 5 2 4 6 1 3 5

8. Sample Problem 1 From the position versus time graph shown, graph the
velocity versus time graph.
Pos(t) t
V(t) t

9. Sample Problem 2 From the position versus time graph shown, graph the
velocity versus time graph.
Here for one moment the slope
is 0. This is a critical value.
Pos(t) t
V(t) t

10. Critical Values
Critical values is a math term for points of a graph that have zero slope.
These points are used for one of two things
A Maximum value
A Minimum value
For example a ball thrown upwards reaches maximum height (position) when its velocity (the slope of a position vs. time graph) is zero.
The maximum height is a critical value

11. Concavity Concavity describes the way the graph curves.
Or, in other words, how the slope is changing
For a position graph the concavity represents the acceleration
Concavity comes in three forms
Positive Concavity, Negative Concavity, and No Concavity

12. The Types of Concavity for position vs. time graphs
Positive concavity
The function curves upwards
The object’s acceleration is +
Negative Concavity
The function curves downwards
The object’s acceleration is -
No Concavity
These are linear functions
The object’s acceleration is 0

13. Speeding up and slowing down
To speed up an object, it’s velocity and acceleration must be the same sign
+ velocity and + acceleration
+ slope and + concavity
-velocity and – acceleration
- slope and - concavity
To slow down an object, it’s velocity and acceleration must be opposite signs
+ velocity and - acceleration
+ slope and - concavity
-velocity and + acceleration
- slope and + concavity

14. Graphs of increasing velocity
Position
Acceleration
+ concavity
- concavity
Velocity
+ slope
- slope
Position

15. Graphs of Decreasing velocity
Position
Acceleration
- concavity
+ concavity
Velocity
+ slope
- slope
Position

16. Role of Area
Area of a graph shows up the built up effect of what is being graphed.
The area of an acceleration vs. time graph is the change in velocity
The area of a velocity vs. time graph is displacement
Areas above the X axis are + changes, while area’s below the X axis are – changes.

17. Velocity vs. time 2 6m 2m
Velocity
(m/sec) 1 2 3
Time (sec)
Position vs. time 6 4
Position
(m) 2 1 2 3
Time (sec)

18. Velocity vs. time Velocity (m/sec) 4.5m Position vs. time 1 2 3 41 2 3 4
Time (sec) 3
Position
(m) 2 1 1 2 3
Time (sec)