1. Basics of graphing motion And studying their slopes S.Caesar

2. In the study of kinematics, we consider a moving object as a particle.
Motion in 1 Dimension v 
In the study of kinematics, we consider a moving
object as a particle.
A particle is a point-like mass having infinitesimal
size and a finite mass.
Motion in One Dimension (2053)

3. = 6 m - 2 m Motion in 1 Dimension Displacement
The displacement of a particle is defined
as its change in position. x
Dx = x - xo
= 6 m - 2 m
= 4 m
(m) -6 -4 -2 2 4 6
Note: Motion to the right is positive
Motion in One Dimension (2053)

4. = -6 m - 6 m Motion in 1 Dimension Displacement
The displacement of a particle is defined
as its change in position. x
Dx = x - xo
= -6 m - 6 m
= -12 m
(m) -6 -4 -2 2 4 6
Note: Motion to the left is negative
Motion in One Dimension (2053)

5. = (2 m) - (-6 m) Motion in 1 Dimension Displacement
The displacement of a particle is defined
as its change in position. x
Dx = x - xo
= (2 m) - (-6 m)
= 8 m
(m) -6 -4 -2 2 4 6
Note: Motion to the right is positive
Motion in One Dimension (2053)

6. Graphs tell a story about motion
We will use 3 different graphs to tell our story:
Position (distance) vs time graph (p-t graph)
Velocity vs time graph (v-t graph)
Acceleration vs time graph (a-t graph)

7. The importance of the SLOPE
The slope of a line on a position-time graph gives you the velocity p
Pick any two points on the line.
Since slope = rise
run
In this case the rise = ∆position
the run = ∆time
Slope = rise = (6m – 2m) = 4m = .5 m/s
run = (12s – 4s) 8s 6m 2m 4s
12s t

8. Velocity is represented displacement-time graph
Motion in 1 Dimension
Average velocity
The average velocity of a particle is defined as x x1 x2 t1 t2 Dx Dt
Velocity is represented
by the slope on a
displacement-time graph t
Motion in One Dimension (2053)

9. Figure 2-24 Graph of position vs
Figure 2-24 Graph of position vs. time for an object moving at a uniform velocity of 11 m/s.

10. Act out these “stories”
Position
time

11. Position
time

12. Position
time

13. Position
time

14. Find the velocities?
Position
Position 20 30 10 10
time
time

15. The average speed of a particle is defined as
Motion in 1 Dimension
Average speed
The average speed of a particle is defined as
Motion in One Dimension (2053)

16. Velocity
In everyday language we use “speed” and “velocity” interchangeably.
In physics they are different.
speed = distance (total length of traveled path)
time (total time of travel)
velocity = displacement
(distance if you would have gone directly there)

17. Displacement The displacement is written: Left:
Displacement is positive.
Right:
Displacement is negative.

18. = 6 m - 2 m Motion in 1 Dimension Displacement
The displacement of a particle is defined
as its change in position. x
Dx = x - xo
= 6 m - 2 m
= 4 m
(m) -6 -4 -2 2 4 6
Note: Motion to the right is positive
Motion in One Dimension (2053)

19. Vector vs Scalar
If someone asks you for directions they are actually asking you for TWO pieces of information….
They want to know:
HOW FAR
WHAT DIRECTION
They are asking you for a vector
A vector has magnitude and direction

20. Vector How do you get to CVS Pharmacy? You go 1.5 miles SOUTH
direction
Magnitude

21. Vector direction
The direction of a vector can be indicated in many ways….
Directions of the compass (ex: South, 600 E of N)
Up, down, left, right
“+”, “-” The direction of “+” has to be indicated
So “ – 10m” would be 10m in the negative direction

22. There are very few straight lines in real life……
What do we do with curves on a
position-time graph?
Look at the following graph and try to figure out what is happening……(teacher slides)

23. Acceleration is represented
Motion in 1 Dimension
Average acceleration
The average acceleration of a particle is defined
as the change in velocity Dvx divided by the time
interval Dt during which that change occurred. v v1 v2 t1 t2 Dv Dt
Acceleration is represented
by the slope on a
velocity-time graph t
Motion in One Dimension (2053)

24. Displacement, velocity and acceleration graphs x
Motion in 1 Dimension
Displacement, velocity
and acceleration graphs x
The slope of a displacement-time
graph represents velocity t v
The slope of a velocity-time
graph represents acceleration t a t
Motion in One Dimension (2053)

25. Displacement, velocity and acceleration graphs x
Motion in 1 Dimension
Displacement, velocity
and acceleration graphs x
The area under a velocity-time
graph represents displacement. Dx t v
The area under an acceleration-time
graph represents change in velocity. Dv t a Dt t
Motion in One Dimension (2053)

26. Definitions of velocity and acceleration
Motion in 1 Dimension
Definitions of velocity and acceleration
Average velocity
Average acceleration
Motion in One Dimension (2053)

27. ? Motion in 1 Dimension For constant acceleration
An object moving with an initial velocity vo undergoes
a constant acceleration a for a time t. Find the final velocity. vo ? a
time = 0
time = t
Solution:
Eq 1
Motion in One Dimension (2053)

28. What are we calculating?t a DV
Motion in One Dimension (2053)

30. Instantaneous Velocity
This means how fast are you going THIS INSTANT IN TIME?
How do you find out this information while you are driving?
Look at your speedometer AND your compass.
How do you find out this information from a graph?
Find the tangent of the curve AT THAT INSTANT.

31. How do you find the tangent on a curve?

32. What does the tangent tell you?
It gives you the slope of the line AT THAT POINT.
The slope of the line on the p-t graph gives you the magnitude of the velocity
The direction of the slope (+ or -) gives you the direction of the velocity
(For those of you in calculus – you can find the “derivative” of the curve at that point to get the tangent)

33. tA

34. Acceleration
Acceleration

35. Name 3 things in your car that can cause acceleration
Gas pedal (sometimes called the “accelerator”!)
Brake
Steering wheel (What???!!)
Any change in direction is also a change in velocity (since it can change direction)

36. So acceleration is any kind of change in velocity.
Going faster
Going slower
Changing direction

37. What does NON-constant acceleration look like?
Don’t panic – we won’t
calculate anything
with changing acceleration!!

38. “Constant” acceleration
In real life
How long does it take your car to go from
0-40 mph ?
How long WOULD it take for your car to go from mph? (that’s ALSO a change of 40 mph in case you weren’t paying attention)
Don’t EVEN think about trying it!
Both the above situation had same acceleration

39. So what do we mean by “Constant Acceleration”
For the sake of simpler math (yeah!) we will assume that the acceleration of something happens at the same rate no matter what.
That would mean that it would take the same amount of time for your car to accelerate from mph as it did for it to accelerate from 0-40 mph.

40. ∆ v = v2 – v1 ∆ v = +10 m/s ∆ v = 0 m/s ∆ v = -10 m/s ∆ v = -10 m/s
before
after
before
after A
∆ v = +10 m/s D
∆ v = 0 m/s
+10 m/s
+20 m/s
+20 m/s
+20 m/s E B
∆ v = -10 m/s
∆ v = -10 m/s
+30 m/s
+20 m/s
+10 m/s C F
∆ v = -20 m/s
∆ v = -10 m/s
+10 m/s
-20 m/s
-10 m/s
-10 m/s
Most Negative _____ _____ _____ _____ _____ _____ Most Positive
C B E F D A

41. If these vectors (lovingly called arrows
If these vectors (lovingly called arrows!) represent motion, what is happening?
Notice that the acceleration vectors are the SAME length. That means the acceleration is CONSTANT! a V1 a V2 V3
So we know the object is speeding up, going in the same direction
How do we represent HOW MUCH it sped up?
By using acceleration vectors

42. - + a V1 a V2 V3 Velocity vectors are positive.
Acceleration vectors are positive.

43. - + a V1 V2 a V3
We know the object is speeding up, in the negative direction.
Velocity vectors are negative.
Acceleration vectors are negative.

44. + - V1 V2 a a V3
We know the object is slowing down, in the negative direction.
Velocity vectors are negative.
Acceleration vectors are positive.

45. Acceleration
There is a difference between “negative acceleration” and “deceleration”:
“Negative acceleration” is acceleration in the negative direction as defined by the coordinate system.
“Deceleration” occurs when the acceleration is opposite in direction to the velocity.

46. + + + - - + Scenario Sign of velocity Sign of acceleration
The driver has stepped on the accelerator and the car is just starting to move forward.
The car is moving forward. The brakes have been applied. The car is slowing down, but has not yet come to rest.
The car is moving backward. The brakes have been applied. The car is slowing down, but has not yet come to rest. + + + - - +

47. So remember…..
When the velocity vectors and the acceleration vectors are in the SAME direction, the object is speeding up.
When the velocity vectors and the acceleration vectors are in the OPPOSITE direction, the object is slowing down.

48. The important question……
How fast does that change in velocity take place?

49. Acceleration Acceleration is the rate of change of velocity.
How fast will the car be going after 1 sec?
How fast will the car be going after 2 seconds?
How fast will the car be going after 5 seconds?

50. What will be the velocity after 5 seconds?
Acceleration
Acceleration is a vector, although in one-dimensional motion we only need the sign.
The previous image shows positive acceleration; here is negative acceleration:
What will be the velocity after 5 seconds?

51. Our new equation…. a = ∆v = v2 – v1 = the change in velocity
∆t ∆t the time it took to change
We abbreviate it as a = ∆v t
The SI unit of velocity is m/s(meter/second)
The SI Unit of time is s (seconds)
∆ -delta means change

52. Back to graphing…… Interactive MINI Demo Things to remember…..
“Cart on a ramp”
Discussion……
Things to remember…..
Accelerating motion will be a curve on a p-t graph
Since we are having a constant acceleration, the a-t graph will always be horizontal
In a v-t graph –constant acceleration has a steady slope (straight line)

53. Motion in 1 Dimension More Graphs More Graphs Practice Session
Consider the situations below and plot graph
More Graphs
Practice Session
Motion in One Dimension (2053)

54. -6 -5 -4 -3 -2 -1 1 1 2 3 4 5 6
Motion in One Dimension (2053)

55. -6 -5 -4 -3 -2 -1 1 1 2 3 4 5 6
Motion in One Dimension (2053)

56. -6 -5 -4 -3 -2 -1 1 1 2 3 4 5 6
Motion in One Dimension (2053)

57. 6 5 4 3 2 1 1 2 4 6 8 10 12 -1 -2 -3 -4 -5 -6
Motion in One Dimension (2053)

58. m 6 5 4 3 2 1 1 2 4 6 8 10 12 -1 s -2 -3 -4 -5 -6
Motion in One Dimension (2053)

59. 6 m 4 2 2 4 6 8 10 12 -2 s -4 -6 2 v 1 t 4 8 12
(s) -1
(m/s) -2 -3
Motion in One Dimension (2053)

60. 6 m 4 2 2 4 6 8 10 12 -2 s -4 -6 2 v
+8 m 1
+4 m t 4 8 12
(s) -1
(m/s)
-12 m -2 -3
Motion in One Dimension (2053)

61. 4 8 12 16 20 24 28
(s) x 4 8 12 16 20 24 28
(m) 1 2 3 t 5
Motion in One Dimension (2053)

62. (s) x 4 8 12 16 20 24 28
(m) 1 2 3 t 5 1 2 3 4 5 t
(s) 6 8 10 v
(m/s)
Displacement
25 m
Motion in One Dimension (2053)

63. (a) Velocity vs. time and (b) displacement vs
(a) Velocity vs. time and (b) displacement vs. time for an object with variable velocity.
You make the v-t graph
for the p-t graph
shown to the right.

64. V1 = 0 (released from rest), origin is to the left, positive is to the right, and no friction
Final location p t v t a t

65. Study more of these type of graphs
Examples of using area under the curve of the v-t graph to determine the displacement.
Study more of these type of graphs

66. Constant velocity v 10 m/s * 4 s = 40 m 10 m/s 10 m/s * 6 s = 60 m 2 s
If an object travels at 10 m/s for 4 seconds, how far did it travel?
10 m/s * 4 s = 40 m
10 m/s
If an object travels at 10 m/s for 6 seconds, how far did it travel?
10 m/s * 6 s = 60 m
2 s
4 s
6 s t

67. Motion with accelerationv
If an object starts from rest and accelerates to 10 m/s in 6 seconds, how far did it travel?
10 m/s
d = ½ (10m/s)(6 s)
d = 30 m
2 s
4 s
6 s t

68. Motion with acceleration and initial velocity ≠ 0
10 m/s
If an object starts from an initial velocity of 5 m/s and accelerates to 10 m/s in 6 seconds, how far did it travel?
5 m/s t
2 s
4 s
6 s
d = area of rectangle + area of triangle
d = (5 m/s)(6 s) + ½ (10 m/s – 5 m/s)(6 s)
d = 30 m + 15 m = 45 m

69. Calculate the total stopping distance
Calculate the total stopping distance. Reaction time for this driver is 0.50s.

70. Figure 2-16 Example 2–9. Graph of v vs. t.

71. What is the displacement of the object from 4s to 6s?v
10 m/s
If an object starts from an initial velocity of 5 m/s and accelerates to 10 m/s in 6 seconds, how far did it travel between seconds 4 and 6?
7.5 m/s
5 m/s t
2 s
4 s
6 s
d = area of triangle + area of rectangle
d = ½ ( m/s)(2 s) + (7.5 m/s)(2 s)
d = 2.5 m + 15 m = 17.5 m

72. A special case of constant acceleration…….
GRAVITY!!

73. Demo……

74. How much gravity is in the tube?
Think about it……..

75. Name 3 ways you can get a piece of paper to fall as fast as a book.
Mini demonstrations are done the students
in groups
Note the student answers-brainstorm session

76. If you drop an object in the absence of air resistance, it accelerates downward at 9.8 m/s2. If instead, you THROW it downward, its downward acceleration AFTER release is:
1. less than 9.8 m/s2
m/s2
3. more than 9.8 m/s2

77. You are throwing a ball straight up in the air
You are throwing a ball straight up in the air. At the highest point, the ball’s ….
1. velocity and acceleration are zero.
2. velocity is nonzero but its acceleration is zero.
3. acceleration is nonzero, but its velocity is zero.
4. velocity and acceleration are both nonzero.

78. Air Resistance
(a) A ball and a light piece of paper are dropped at the same time. (b) Repeated, with the paper wadded up.

79. Study the direction of the velocity and acceleration as the object is thrown up Study the direction of the velocity and acceleration as the object is falling down.
An object thrown into the air leaves the thrower's hand at A, reaches its maximum height at B, and returns to the original position at C. Examples 2-12, 2–13, 2–14, and 2–15.

80. Figure 2-30 Problem 11.

81. Figure 2-40 Problem 73.

82. Problem Solving Skills
1. Read the problem carefully
2. Sketch the problem
3. Visualize the physical situation
4. Strategize
5. Identify appropriate equations
6. Solve the equations
7. Check your answers
Motion in One Dimension (2053)