1. Chapter 5 A Mathematical Model of Motion
Physics
Chapter 5
A Mathematical Model of Motion

2. A Mathematical Model of Motion
5.1 Graphing Motion in One Dimension
5.2 Graphing Velocity in One Dimension
5.3 Acceleration
5.4 Free Fall

3. Graphing Motion in One Dimension
Motion can be described in words, sketches, diagrams, graphs and equations
Graphs of motion can be read to determine various characteristics of motion
Motion graphs have time on the independent (x) axis and either position, velocity or acceleration on the dependent (y) axis
Reading these graphs can give you an object’s position, velocity and/or acceleration at any time during its motion

4. Position-Time Graphs
These graphs show an object’s position at any time (instant)
Position—where an object is located relative to a reference point
Y (position) versus X (time)

5. Position-Time Graphs
What does this graph tell you about the object’s motion?

6. Position-Time Graphs
This graph tell you about the object’s position and it also tells you about its speed (velocity)
Velocity—the rate of change of an object’s position (displacement) for a time interval

7. Position-Time Graphs
Slope of a position-time graph gives you the average velocity of the object
Slope = rise/run = y/x
= (d1- do)/(t1- to)
Slope = meters/seconds = average velocity

8. Position-Time Graphs
What does this graph tell you about the object’s motion?

9. Position-Time Graphs
What does this graph tell you about the object’s motion?

10. Position-Time Graphs
What does this graph tell you about the object’s motion?

11. Position-Time Graphs How are the graphs below different?
What do they tell you about the object’s motion?

12. Position-Time Graphs
What does this graph tell you about the object’s motion?

13. Position-Time Graphs
So far, all the graphs have been of uniform motion—equal displacements occur over equal successive time intervals
Let’s see when this is not the case

14. Position-Time Graphs
What does this graph tell you about the object’s motion?

15. Position-Time Graphs
What does this graph tell you about the object’s motion?

16. Position-Time Graphs
For this graph we can find the instantaneous velocity of the object at any time
But without calculus, we can only find the slope of a straight line
To to this we must draw a straight line called a tangent line
Tangent line—a line that hits a curve at only one point and doesn’t cross the line
The slope of this tangent line will give use the velocity of the object at that instant

17. Position-Time Graphs
At 6 seconds we draw a tangent line and then find the slope of that line to find the velocity at that time only (instantaneous velocity)

18. Position-Time Graphs Slope = rise/run = y/x = (100-20) (11-5)
= (100-20) (11-5)
= 13.3 m/s
at 6 s

19. v = d/t = (d1 – do)/(t1 – to)
Velocity Equations
Uniform motion can be represented by algebraic equations
Average Velocity
v = d/t = (d1 – do)/(t1 – to)

20. Position Equations v = (d1 – do)/t1 d1 = do + vt
Since most motion starts at to = 0
v = (d1 – do)/t1 or
d1 = do + vt

21. Graphing Motion in One Dimension
Summary
Uniform motion
Equal displacements during equal time intervals
Position-Time graph
Reading it tells you an object’s position at any time
Slope of line (curve) gives you the object’s velocity (average or instantaneous)
Position equation
d1 = do + vt

22. Graphing Velocity in One Dimension
A velocity-time graph gives you an object’s velocity at any time
This graph is my favorite graph —reading it gives you the most motion information of any graph!

23. Graphing Velocity in One Dimension
What does this graph tell you about an object’s motion?

24. Graphing Velocity in One Dimension
What does this graph tell you about an object’s motion?

25. Graphing Velocity in One Dimension
What does this graph tell you about an object’s motion?

26. Graphing Velocity in One Dimension
In which graph is the object moving faster?

27. Graphing Velocity in One Dimension
What does this graph tell you about an object’s motion?

28. Graphing Velocity in One Dimension
Is the velocity constant in this graph?
How do you know?

29. Graphing Velocity in One Dimension
The slope of a P-T graph gives you an object’s velocity
The slope of a Velocity-time graph gives you an object’s acceleration
Acceleration—the rate of change in an object’s velocity during a time interval

30. Graphing Velocity in One Dimension
Slope = rise/run = v/t = (m/s)/s = m/s2
a = (v1 – vo)/(t1 – to)

31. Graphing Velocity in One Dimension
You can also find an object’s displacement from a Velocity-Time graph
Area = b x h (or ½ bh)
Area = (m/s)(s)
Area = meters

32. Graphing Velocity in One Dimension
What does this graph tell you about an object’s motion?

33. Graphing Velocity in One Dimension
Find the velocity at 4 seconds
Find the average acceleration and displacement between 0 and 4 seconds

34. Graphing Velocity in One Dimension
Summary
Velocity-Time Graph
Reading the graph gives you an object’s velocity at any time
Slope of V-T graph gives you an object’s acceleration
Area under the curve of a V-T graph gives you an object’s displacement

35. Acceleration Acceleration—the rate of change of velocity
An acceleration-Time graph gives an object’s acceleration rate at any time

36. Acceleration
What does this graph tell you about the object’s motion?

37. Acceleration
What does this graph tell you about the object’s motion?

38. Acceleration
What does this graph tell you about the object’s motion?

39. Acceleration
The area under the curve for an acceleration-time graph gives you an object’s change in velocity
Area = b x h = (m/s2)(s) = m/s

40. Acceleration Find the acceleration rate at 2 seconds.
Find the change in velocity of the object between 2 and 6 seconds

41. Acceleration
What does this graph tell you about the object’s motion?

42. Acceleration Summary Acceleration-Time Graph
Reading the graph gives you an object’s acceleration at any time
Area under the curve of a A-T graph gives you an object’s change in velocity

43. Graph Summary Graph Read it Slope it Area it Position-Time (P-T)
Position at any time
Average velocity
Garbage
Velocity-Time (V-T)
Velocity at any time
Average acceleration
Displacement
Acceleration-Time (A-T)
Acceleration at any time
Junk
Change in velocity

44. Equations of Motion for Uniform Acceleration
d1 = do + v t
v1 = vo + āt
d1 = do + vot + 1/2āt2
v1 2 = v02 + 2ā(d1 -do)
d1 = do + ½(v1 + vo)t
Also….
v = 1/2(v1 + vo)
v = d/t
ā = v/t

45. Equations of Motion for Uniform Acceleration
Click here for Chapter 5 Problem Solving: New Equations

46. Free Fall
When you drop a feather and a rock from the same height, which one hits the ground first?
Why?
Book and paper demo

47. Free Fall Do heavier objects fall faster than lighter objects?
How can I make a piece of paper fall faster/slower?
Which has more air resistance; an elephant or a snowflake?
Why doesn’t an elephant fall slower than a snowflake?

48. Free Fall
Actually without air resistance (in a vacuum) all objects fall at the same rate due to gravity
Acceleration due to gravity (g) = m/s2
On Earth
Downward (-)

49. Free Fall Acceleration due to gravity (g) = - 9.8 m/s2
This means that as any object falls its velocity increases by m/s each second.
An object that moves through the air only under the influence of gravity is said to be in free fall.

50. Free Fall
If an object is dropped, each second its downward velocity increases by 9.8 m/s
After 1 sec; m/s
After 2 sec; m/s
After 3 sec; m/s
After 4 sec ????

51. Free Fall
If an object is thrown upward, each second its downward velocity increases by 9.8 m/s
If it starts with 19.6 m/s of upward velocity
After 1 sec; loses 9.8 m/s, so is going 9.8 m/s
After 2 sec; loses another 9.8 m/s, so stops (0 m/s)
After 3 sec; gains -9.8m/s, so is going –9.8 m/s
After 4 sec ????