1. Graphical Analysis of Motion
AP Physics C

2. Slope – A basic graph model
A basic model for understanding graphs in physics is SLOPE.
Using the model - Look at the formula for velocity.
Who gets to play the role of the slope? Velocity
Who gets to play the role of the y-axis or the rise? Displacement
Who get to play the role of the x-axis or the run? Time

3. Displacement vs. Time graph
What is the velocity of the object from 0 seconds to 3 seconds?
V=∆x/t= 30 m- 0 m/(3s) = 10 m/s

4. Displacement vs. Time graph
What is the velocity of the object from 7 seconds to 8 seconds?  Once again...find the slope!
v= ∆x/t = 0m/s

5. Displacement vs. Time graph
What is the velocity from 8-10 seconds? You must remember! To find the change it is final - initial.
v=∆x/t = -90m/2s = -45 m/s

6. Example
It is very important that you are able to look at a graph and explain it's motion in great detail. These graphs can be very conceptual.
Look at the time interval t = 0 to t = 9 seconds. What does the slope do?
Object accelerates and speeds up while moving to the right.
Look at the time interval t = 9 to t = 11 seconds. What does the slope do?
Object is at rest.
Look at the time interval t = 11 to t = 15 seconds. What does the slope do?
Object moves at constant velocity while moving to the right.
Look at the time interval t = 15 to t = 17 seconds. What does the slope do?
Object moves at constant velocity while moving to the left.

7. Slope – A basic graph model
Let’s look at another model
Who gets to play the role of the slope? Acceleration
Who gets to play the role of the y-axis or the rise? Velocity
Who get to play the role of the x-axis or the run? Time 
What does all the mean? It means that if your are given a Velocity vs. Time graph. To find the acceleration of an object during specific time intervals simply find the slope.

8. Velocity vs. Time Graph What is the acceleration from 0 to 6s?
a=∆v/t= 60 m/s / 6s = 10 m/s2
What is the acceleration from 6 to 9s?
a=∆v/t = 0 m/s/ 3s = 0 m/s2
You could say one of two things here:
Object does not accelerate or object moves at constant velocity.
What is the acceleration from 14 to 15s?
a=∆v/t=-10 m/s/ 1s = -10m/s2
A negative acceleration is sometimes called DECELERATION. Why is this not entirely accurate? Negative acceleration can mean that the object speeds up while moving in the negative direction or slows down while moving in the positive acceleration. Only the second scenario involves deceleration/

9. Velocity vs. Time Graph
Conceptually speaking, what is the object doing during the time interval t = 9 to t = 13 seconds?
Does the steepness or slope increase or decrease? The steepness decreases so the acceleration must be decreasing as well.

10. Area – the “other” basic graph model
Another basic model for understanding graphs in physics is AREA.
Let's try to algebraically make our formulas look like the one above. We'll start with our formula for velocity.
d=vt
Who gets to play the role of the base? Time
Who gets to play the role of the height? Velocity
What kind of graph is this? Velocity vs Time
Who gets to play the role of the Area?
Displacement

11. Example
What is the displacement during the time interval t = 0 to t = 5 seconds?
What is the displacement during the time interval t = 8 to t = 12 seconds?
During this time period we have a triangle AND a square. We must find the area of each section then ADD them together.
87. 5m m

12. Area – the “other” basic graph model
Let's use our  new model again, but for our equation for acceleration.
What does this mean? Who gets to play the role of the base?
Who gets to play the role of the height?
What kind of graph is this?
Who gets to play the role of the Area?
Time
Acceleration
An Acceleration vs. Time graph  ( acceleration = y-axis & time = x-axis)
The Velocity

13. Acceleration vs. Time Graph
What is the velocity during the time interval t = 3 and t = 6 seconds?
(∆v) = at
∆v=at = 6m/s2 (3s) = 18 m/s
18 m/s

14. Summary There are 3 types of MOTION graphs
Displacement(position) vs. Time
Velocity vs. Time
Acceleration vs. Time
There are 2 basic graph models
Slope
Area

15. Summary v (m/s) a (m/s/s) x (m) t (s) t (s) t (s) slope = v slope = a
area = v
area = x
t (s)
t (s)
t (s)

16. Comparing and Sketching graphs
One of the more difficult applications of graphs in physics is when given a certain type of graph and asked to draw a different type of graph
x (m)
slope = v
t (s)
v (m/s)
How could you translate what the SLOPE is doing on the graph ABOVE to the Y axis on the graph to the right?
t (s)

17. Example v (m/s) x (m) 1st line constant velocity left moving 2nd Line
t (s)
t (s)
1st line
constant velocity
left moving
2nd Line
Object is at rest
3rd line
Object moves at constant velocity
Moves to right

18. Example – Graph Matching
What is the SLOPE(a) doing?
a (m/s/s)
v (m/s)
t (s)
a (m/s/s)
t (s)
a (m/s/s)
t (s)
t (s)

19. V-t Graph Problem 1 Calculate the following: Acceleration from 0 -10s
Distance traveled from s
Displacement from 15-40s
Acceleration from 40-55s
Displacement after 55s
Draw the corresponding a-t graph
6 m/s m 500 m 2.67 m/s2

20. V-t graph problem 2 Draw a corresponding a-t graph. Find the distance
Traveled from 0-10s

22. 1. Match each of the following v-t graphs with the corresponding a-t graph
a-e b-d c-f

23. 2. The following graph shows the position of a roller coaster as it changes with time.
Explain the object’s motion over each interval.
Speeds up while moving right b) Speeds up, hits constant velocity, and slows down while moving right. c) Moves at constant velocity to the right
d) Speeds up, hits constant velocity, and then slows down while moving right. e) Moves at constant velocity to the right.

24. 3. A particle has the acceleration versus time graph shown below: Draw the corresponding v-t and x-t graphs.