1. Graphing Motion

2. Before we begin though…..a few things
When we look at a line with no numbers we are just looking at the shape, analyzing the shape of the line. Slope can help with that.

3. Before we begin though…..a few things
The x axis is always time, the y axis is either labeled position, velocity, or acceleration.
The y axis is determined by the type of question

4. Sometimes you will a graph in these formsx v t t a t

5. Before we begin though…..a few things
When we analyze points using slope we are finding the average velocity
Average velocity – the average of all instantaneous velocities; found simply by a distance/time ratio.
Instantaneous velocity -  the velocity at any given instant in time. (the velocity at one specific point)
Average velocity is different than instantaneous velocity
The same principle is applied with acceleration
Avg acceleration – average of all accelerations
Instantaneous acceleration – acceleration at one specific point

6. There are three main types of graphs in Kinematics.
Displacement vs. time graphs.
Velocity vs. time graphs.
Acceleration vs. time
***Kinematics is the science of describing the motion of objects using words, diagrams, numbers, graphs, and equations
Let’s look at the displacement vs. time graph.

7. displacement vs. time graph
aka
position vs. time graph x t

8. From any position x time graph we can determine the average velocity.
The slope of the line will give you the average velocity.
Slope =
rise
run =
∆ in vertical coordinates
∆ in horizontal coordinates
Imagine that you are running a marathon and we have decided to graph your movement.

9. Slope = d/t = v
0-90s. – the slope of d/t graph always equals the velocity of that object during that time interval.
Slope = d/t = v
Since it is moving up in a straight line you have a positive constant velocity.

10. 90s-150s – during this time your position has stayed the same
90s-150s – during this time your position has stayed the same. So v = 0m/s

11. 150s-240s – again finding the slope will give you your velocity.
In this section you are moving at a faster constant velocity.

12. 240s-300s – again finding the slope will give you your velocity
240s-300s – again finding the slope will give you your velocity. But notice it is negative. This means you turned around.
So your velocity is negative.

13. 300s-360s – again the line is horizontal.
You must have stopped.

14. 360s-510s – the line curves upward.
This means the slope of the line is getting larger.
Since slope = velocity you are accelerating!

15. Just a quick tip on acceleration on a displacement vs. time graph
● If you see any part of the happy clown's face on a graph, it is positive acceleration.
● If you see any part of the sad clown's face, it is negative acceleration.

16. One last thing about displacement graphs.
When the line reaches zero “0” on the x-axis.
The object is said to be “at rest”

17. velocity vs. time graph v t

18. Velocity vs. time graphs.
First thing to remember.
The rules are not the same for both displacement vs time and velocity vs time graphs!
The lines mean different things.
For the next example, the graph is from the same data (the marathon)

19. 0s-90s – you were running at a positive constant velocity
0s-90s – you were running at a positive constant velocity. On a v vs t graph, a flat line means constant velocity.

20. 90s-150s – You stopped, notice that “stopped” is shown by a horizontal line exactly at 0m/s.
It is a flat line at zero, which means your velocity was zero.

21. 150s-240s – You are running forward again
150s-240s – You are running forward again. To show a faster velocity than earlier, we have a flat line that is higher than the previous one.

22. 240s-300s – This is when you ran back to a previous area.
You are running at -3.3m/s

23. 300s-360s – Again, horizontal line at zero. You must have stopped.

24. 360s-510s – On a v/t graph the line does not curve and has a positive slope.
The slope is equal to the acceleration: ∆V
Slope =
= a t

25. acceleration vs. time graph

26. Acceleration vs. Time Graphs.
In the v vs. t graph, we showed that the slope of the line showed an increase (change) in velocity over time.
∆v / t = acceleration.
Because this change in velocity over time is a constant change in velocity, that means the acceleration itself is also constant.

27. Acceleration vs. Time Graphs.
Acceleration that is constant is shown on a graph by a flat straight line.

28. Acceleration vs. time – how to graph it.X
a vs. t graph
Straight line in positive direction means the acceleration is also constant over time.
v vs. t graph
Straight line in positive direction means it is increasing at a constant velocity..
x vs. t graph
Displacement getting more positive (greater) over time.
If you graphed this as v vs. t it would be…

29. v vs. t graph a vs. t graph x vs. t graph
has a constant decrease of velocity over
time which is called deceleration.
a vs. t graph
there is a constant negative acceleration or deceleration in the graph X
x vs. t graph
shows a graph of an object moving backwards at a decelerating speed.

30. v vs. t graph
Since it is moving toward the point of origin at a positive constant velocity we can graph it with a line like this.
a vs. t graph
there is a constant positive acceleration in the graph
x vs. t graph
here the object is moving at
an accelerating speed towards the point of origin.