1. Graphing Motion

2. Graphing Motion in One Dimension
Interpret graphs of position versus time for a moving object to determine the velocity of the object
Describe in words the information presented in graphs and draw graphs from descriptions of motion
Write equations that describe the position of an object moving at constant velocity

3. Parts of a Graph
X-axis
Y-axis
All axes must be labeled with appropriate units, and values.

4. Position vs. Time The x-axis is always “time”
The y-axis is always “position”
The slope of the line indicates the velocity of the object.
Slope = (y2-y1)/(x2-x1)
d1-d0/t1-t0
Δd/Δt

5. Uniform Motion
Uniform motion is when the velocity of an object does not change
Straight lines on position-time graphs mean uniform motion.

6. Given below is a diagram of a ball rolling along a table
Given below is a diagram of a ball rolling along a table. Strobe pictures reveal the position of the object at regular intervals of time, in this case, once each 0.1 seconds.
Notice that the ball covers an equal distance between flashes. Let's assume this distance equals 20 cm and display the ball's behavior on a graph plotting its x-position versus time.

7. The slope of this line would equal 20 cm divided by 0
The slope of this line would equal 20 cm divided by 0.1 sec or 200 cm/sec. This represents the ball's average velocity as it moves across the table. Since the ball is moving in a positive direction its velocity is positive. That is, the ball's velocity is a vector quantity possessing both magnitude (200 cm/sec) and direction (positive).

8. Steepness of slope on Position-Time graph
Slope is related to velocity
Steep slope = higher velocity
Shallow slope = less velocity

9. Different Position. Vs. Time graphs
Accelerated Motion
Uniform Motion
Constant positive velocity
(zero acceleration)
Increasing positive velocity
(positive acceleration)
Constant negative velocity
(zero acceleration)
Decreasing negative velocity
(positive acceleration)

10. Different Position. Vs. Time
Changing slope means changing velocity!!!!!!
Increasing negative slope = ??
Decreasing negative slope = ??

11. 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

12. 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) 2-5 s, 6-7 s B) 9-11 s C) 0-2 s D)7-9 s E) 6-7 s and 9-11 s, also 0-2 s and 5-6 s and 7-9 s

13. 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

14. Tangent Lines t On a position vs. time graph: SLOPE VELOCITY Positivex
Tangent Lines t
On a position vs. time graph:
SLOPE
VELOCITY
Positive
Negative
Zero
SLOPE
SPEED
Steep
Fast
Gentle
Slow
Flat
Zero

15. Increasing & Decreasingt x
Increasing
Decreasing
On a position vs. time graph:
Increasing means moving forward (positive direction).
Decreasing means moving backwards (negative direction).

16. Concavity On a position vs. time graph:x
Concavity
On a position vs. time graph:
Concave up means positive acceleration.
Concave down means negative acceleration.

17. Graphing Velocity in One Dimension
Determine, from a graph of velocity versus time, the velocity of an object at a specific time
Interpret a v-t graph to find the time at which an object has a specific velocity

18. Velocity vs. Time X-axis is the “time” Y-axis is the “velocity”
Slope of the line = the acceleration

19. Different Velocity-time graphs

20. Different Velocity-time graphs

21. Velocity vs. Time Horizontal lines = constant velocity
Sloped line = changing velocity
Steeper = greater change in velocity per second
Negative slope = deceleration

22. Acceleration vs. Time Time is on the x-axis
Acceleration is on the y-axis
Shows how acceleration changes over a period of time.
Often a horizontal line.

23. All 3 Graphs t x v t a t

24. Real life
Note how the v graph is pointy and the a graph skips. In real life, the blue points would be smooth curves and the orange segments would be connected. In our class, however, we’ll only deal with constant acceleration. v t a t

25. Constant Rightward Velocity

26. Constant Leftward Velocity

27. Constant Rightward Acceleration

28. Constant Leftward Acceleration

29. Leftward Velocity with Rightward Acceleration

30. Graph Practice Try making all three graphs for the following scenario:
1. Newberry starts out north of home. At time zero he’s driving a cement mixer south very fast at a constant speed.
2. He accidentally runs over an innocent moose crossing the road, so he slows to a stop to check on the poor moose.
3. He pauses for a while until he determines the moose is squashed flat and deader than a doornail.
4. Fleeing the scene of the crime, Newberry takes off again in the same direction, speeding up quickly.
5. When his conscience gets the better of him, he slows, turns around, and returns to the crash site.