1. Mathematical Model of Motion
Ch. 5 Pg

2. 5.1: Graphing Motion in 1-Dimension
Objectives: Today you will……
Interpret graphs of position vs. Time for a moving object to deterimine the velocity of an object
Describe in words the information presented in the graphs and draw graphs from descriptions of motion
Write equations that describe the position of an object moving at constant velocity.

3. Position-Time Graphs
x/y plot or graph that relates the position of an object at given time intervals.
Most commonly we set our origin at t=0, but the horizontal origin if up to you.
Consider the motion of a running back:
Running back takes a handoff at the 10 yard-line and is finally tackled at the 48 yard-line
The x position intervals for 6 seconds are: 10, 12, 18, 26, 36, 43, and 48
Create a motion diagram, using vectors first
Then graph the running back’s position vs time using an X/Y plot.

4. Position-Time Graphs What does your motion diagram look like?
What does your graph look like?
Does your graph show the exact path of the running back?
Discuss

5. What is an instant of time?
How do we define an instant of time?
How long does it last?
An instant of time is a snapshot that lasts 0 seconds.
How long was running back at each position we graphed?
The x-points of the running back are simple snapshots of where the running back was horizontally at any instant in time.

6. Assignment
Read Section 5.1 Pg
Will continue notes tomorrow.

7. Using P/T Graph to find “when” and “where”
Position time graphs are useful in determining where an object is, or when did something happen to an object.
“when did the running back reach the 30 yard- line?” “Where was the running back after 4.5 seconds?”
Look at Example Problem on pg. 83 of textbook

8. Graphing the Motion of 2 or More Objects
We can also use a position-time graph to compare multiple objects.
Look back at the running back problem.
Now lets add a middle linebacker (who misses a tackle) and a free safety (who makes the tackle)
Look at Figure 5-3 on pg. 84
Where did each player have the earliest chance to make a tackle?

9. From Graphs to Words and Back
Interpreting a position-time graph:
Locate position of the object at time zero (t=0).
Figure out a general trend and decide if the objects position increases, decreases, etc
Relate motion vs origin and decide positive and negative velocities of the motion.
Motion away from origin is positive or negative. No change in motion would be no velocity
Look at Example Problem on pg. 85
Do practice problems 1-3 on pg. 85

10. Uniform Motion
Uniform motion – equal displacements occur over successive time intervals.
These allow for motion diagrams and position – time intervals can both be used to describe the motion of an object.
Uniform motion follows the rules for linear equations:
Has a slope
To find this on a position time graph we use change in displacement or velocity divided by change in time.
Try Practice Problems 4-8 on pg. 87 of your textbook.

11. Using Equations to find out When and Where
The general equation for velocity is:
V = Δd/Δt or d1-d0/t1-t0
If we rearrange the equation and assume that t0 = 0 then the equation becomes:
D1 = d0 + vt1 and to even further simplify it
D = di + vt
Sample Problem on pg. 88
Try Practice Problems 9-12 on pg. 89 of text

12. Assignment
5.1 Review
Pg. 89
Questions 1 – 4
Due tomorrow.

13. Graphing Velocity in 1-D
Objectives
Determine from a graph of velocity versus time, the velocity of an object at a specific time
Interpret velocity-time graph to find the time at which an object has a specific velocity
Calculate the displacement of an object from the area under the velocity-time curve.

14. Determining Instantaneous Velocity
What does the graph of an accelerating object look like?
Obviously the object is going faster and faster
The motion cannot be uniform because the Δd/Δt are getting continually larger.
Instantaneous velocity does not equal average velocity.
Look at Figure 5-9 on pg. 90 of your textbook

15. Determining Instantaneous Velocity
What was the plane’s velocity at t=1 and t=2?
What is the plane’s average velocity during that time interval?
What is the plane’s instantaneous velocity at 2.5s?
By Reducing the time interval we arrive at an answer closer to the instantaneous velocity of the object

16. Velocity-Time Graphs
What would a velocity-graph look like for an object traveling at 70m/s in 5 seconds?
What about one for an object that goes from 65m/s to 82m/s in 5 seconds?
What about an object traveling at 70m/s in the opposite direction?
If we graphed all 3 line on the same graph, which ones would cross?
What would them crossing tell us?

17. Displacement from a Velocity-Time Graph
For an object at a constant velocity: Δd = vΔt
The change is displament (distance) is equal to the velocity multiplied by the time traveled.
Look at Example Problem on pg. 92 of textbook.
Do Practice Problems on. Pg. 93

18. Assignment
5.2 Review
Pg. 93
Questions 1-5
Due tomorrow

19. 5.3: Acceleration Objectives
Determine from curves on a V-T graph both the constant and instantaneous acceleration
Determine the sign of the acceleration using a V-T graph and motion diagram
Calculate the velocity and displacement of an object undergoing constant acceleration.

20. Determining Avg. Acceleration
Avg. acceleration is the rate of change of velocity between 2 periods of time.
A=Δv/Δt
Unit is meters/ second*second or m/s2
You can find acceleration by using a velocity-time graph
If a vehicle goes from 8m/s to 14 m/s in 2 seconds. What is its average acceleration?

21. Constant and Inst. Acceleration
Constant acceleration – uniform motion of acceleration.
Represented by a constant slope on a V-T graph.
Instantaneous acceleration – the acceleration of an object at an instant of time.
Represented as a
Look at Example Problem on pg. 95

22. Pos. and Neg. Acceleration
Negative acceleration = deceleration
Positive acceleration – object increasing in velocity.
Look at example problem on pg. 96 of text
Try Practice Problems on pg. 97 of text

23. Acceleration when inst. Vel. = 0
Remember that the velocity of any object can be 0 but only for an instant of time.
Remember an instant of time lasts 0 seconds.
Consider a ball being rolled up hill and as it slows down and begins to roll back down hill. What is the component of its acceleration at the instant is has no velocity??

24. Calculating Velocity from Acceleration
a = Δv/Δt or v1-v0/ t1-t0
Assume t0 = 0 and rearrange to v = v0 + at
This can be used to solve for a specific velocity when given acceleration and a time interval
Try Practice Problems on pg. 98

25. Displacement Under Const. Acceleration
Using a velocity time graph can allow us to use area under the curve to figure out displacements when velocity is constant.
Changing the velocity from constant to an accelerating velocity can also provide a displacement:
Several equations can be derived based on conditions:
Remember d = vit add to this the acceleration which is ½(vf +vi)t
This means total displacement will equal:
df= vit + ½ (vf -vi)t or if we combine them
df = ½ (vf+vi)t

26. Displacement Under Const. Acceleration
Now we have to add to this an instance where d0 is not equal to zero. This gives us
df= di + ½(vf+vi)t ; this works if we know the velocity and time
If we do not know velocity, then we have to substitute in the equation for velocity (vi +at); plug this into our derived equations and we get:
df = di + ½(vi + vi + at)t or when combined correctly
df = di + vit +1/2at2
Isn’t physics fun!!!

27. Displacement Under Constant Acceleration
All the derived equations involve different variables. Some use position, time, and velocity; others use position, acceleration, and time. Is there one that uses acceleration, position, and velocity, but not time? You bet!!!
Start with df = di + ½(vf+vi)t and vf=vi + at
Solve second for t: t= (vf-vi)a then substitute
df = di + ½(vi +vf)(vf-vi)/a
Can be rearranged to solve for final velocity as:
Vf2 = vi2 + 2a(df-di)
To make this easier Look at Table 5-2 on p.101 of text

28. Practice Problems and Assignment
Pg. 103 of textbook
27-30
Homework: 5.3 Review
Pg. 103
1-5

29. 5.4: Free Fall Objectives:
Recognize the meaning of acceleration due to gravity
Define the magnitude of the acceleration due to gravity as a positive quantity and determine the sign of the acceleration relative to the chosen coordinate system
Use the motion equations to solve problems involving freely falling objects.

30. Gravitational Acceleration
Gravity has been a known force since the 1600’s
Galileo noticed that objects always fall at the same velocity (ignoring the resistance of the medium they fall through)
This acceleration was measured at 9.8 m/s2 and given the symbol “g”
There will be small variations of “g” at different places on Earth but 9.8 is the average value
“g” is always a positive value.

31. Gravitational Acceleration
The acceleration due to gravity is defined as the acceleration of an object in free fall that results from the influence of gravity.
Remember gravity is acceleration.
This means things acceleration 9.8m/s2
1 second = 9.8
2 seconds = 19.6
3 seconds = 29.4
What about objects thrown upward?
Relies on how we set our origin
Any object thrown upward, when the upward from origin is positive, will always factor in an acceleration of “–g”
This tells us the the velocity of the ball becomes less and less as it travels upward, then will increase in velocity as it falls downward.

32. Gravitational Acceleration
We still use the same equations that we used from the previous sections. We simply use “g” as our acceleration.
Look at Example Problem on pg.105 in textbook
Try Practice Problems on pg. 106 of the textbook

33. Assignment
5.4 Review
Pg. 106
Questions 1 and 2