1. AP Physics C

2. Dimensionality
Dimensionality is an abstract concept closely related to units
Units describe certain types of quantities.
Feet, inches, meters, nanometer - Units of Length
We can develop a set of rules that allow us to:
Check equations
Determine the dependence on specified set of quantities

3. Dimensionality There are 3 types of quantities we will discuss today:
Length
Time
Mass

4. Notation
We denote these quantities as:
Length - L
Time - T
Mass - M

5. Notation
When denoting the dimensionality of a variable we use square brackets [ ]
𝑥 𝑡 =𝑣 𝑡+ 𝑥 0
𝑥 =L
𝑡 =T
𝑣 = L T

6. Rules of Dimensionality
Variables on opposite sides of an equals sign must have the same dimensionality
Variables on opposite sides of a + or - must have the same dimensionality

7. Rules of Dimensionality
Lets check the formula:
𝑥 𝑡 =𝑣 𝑡+ 𝑥 0

8. Rules of Dimensionality
Pure number (2, 3 , 𝜋) are always dimensionless
Special functions (sine, cosine, exponential, etc.) are always dimensionless
The argument of special functions are always dimensionless

9. Dimensional Analysis 𝐸= 1 2 𝑚 𝑣 2
Use the rules of dimensionality to find the dimensions of 𝐸
𝐸= 1 2 𝑚 𝑣 2

10. Dimensional Analysis 𝐸=ℎ 𝑓 𝐸 =M L 2 T 2 ; 𝑓 = 1 T
What are the dimensions of ℎ?
ℎ =?

11. Dimensional Analysis
𝑥 𝑡 =𝐴 sin (𝜔 𝑡)
[𝑥] = L
𝐴 = ?
𝜔 =?

12. Dimensional Analysis Consider a mass swinging on the end of a string.
The period is the amount of time
takes for the mass to complete one
full oscillation
What variables do you
suspect the period of the motion will depend on?

13. 𝑇= 𝑚 𝛼 𝐿 𝛽 𝑔 𝛾 Dimensional Analysis
In general we may assume 𝑇 is some combination of 𝑚, 𝐿, and 𝑔:
𝑇= 𝑚 𝛼 𝐿 𝛽 𝑔 𝛾
Using dimensional considerations, we can solve for 𝛼, 𝛽, and 𝛾

14. Position, Velocity, & Acceleration
In Physics we often need to relate position, velocity, & acceleration
A mathematical description of this relationship requires calculus
In this section we will discuss the graphical relationship between a position vs. time graph and a velocity vs. time graph

15. Graphical Analysis Recall that: 𝑠𝑙𝑜𝑝𝑒= Δ𝑦 Δ𝑥 = 𝑦 𝑓 − 𝑦 0 𝑥 𝑓 − 𝑥 0 𝑦
𝑠𝑙𝑜𝑝𝑒= Δ𝑦 Δ𝑥 = 𝑦 𝑓 − 𝑦 0 𝑥 𝑓 − 𝑥 0 𝑦 Δ𝑦 𝑥 Δ𝑥

16. Graphical Analysis
In Physics we have:
𝑠𝑙𝑜𝑝𝑒= 𝑥 Δ𝑥 𝑡 Δ𝑡

17. Graphical Analysis For a position vs. time graph: 𝑠𝑙𝑜𝑝𝑒= Δ𝑥 Δ𝑡 = 𝑣 𝑎𝑣𝑔
For an velocity vs. time graph:
𝑠𝑙𝑜𝑝𝑒= Δ𝑣 Δ𝑡 =
𝑣 𝑎𝑣𝑔
𝑎 𝑎𝑣𝑔

18. Graphical Analysis 𝑥
𝑣 𝑎𝑣𝑔 = Δ𝑥 Δ𝑡 Δ𝑥 𝑡 Δ𝑡

19. Graphical Analysis 𝑥
𝑣 𝑎𝑣𝑔,1 < 𝑣 𝑎𝑣𝑔,2
Δ 𝑥 2
Δ 𝑥 1 𝑡
Δ 𝑡 1
Δ 𝑡 2

20. Graphical Analysis 𝑥 𝑠𝑙𝑜𝑝𝑒=𝑣(𝑡) 𝑡 𝑡
The slope at one point is the instantaneous velocity.

21. Graphical Analysis 𝑣(𝑡) is the slope of a line tangent to 𝑥(𝑡) at 𝑡
𝑣(𝑡) is graphically understood as the steepness of the 𝑥(𝑡) vs 𝑡 graph.

22. What does 𝑣(𝑡) look like?
Graphical Analysis 𝑥 𝑡
What does 𝑣(𝑡) look like?

23. 𝑥 𝑡 𝑣 𝑡

24. Identify where 𝑣(𝑡) positive, negative, & zero
Graphical Analysis 𝑥 𝑡
Identify where 𝑣(𝑡) positive, negative, & zero

25. 𝑥 𝑡 𝑣 𝑡

26. Graphical Analysis 𝑣 𝑡 𝑡
Sketch a graph of 𝑎(𝑡)

27. 𝑣 𝑡 𝑣 𝑡

28. The Derivative
We can approximate 𝑣 𝑡 0 as the average velocity over a time an interval Δ𝑡 starting at 𝑡 0
𝑣 𝑡 0 ≈ Δ𝑥 Δ𝑡 = 𝑥 𝑡 0 +Δ𝑡 −𝑥 𝑡 0 Δ𝑡

29. The Derivative 𝑥 𝑡
𝑡 0

30. The Derivative 𝑥 𝑡 Δ𝑡

31. The Derivative 𝑥 𝑡 Δ𝑡

32. The Derivative 𝑥 𝑡 Δ𝑡

33. The Derivative 𝑥 𝑡 Δ𝑡

34. The Derivative 𝑥 𝑡 Δ𝑡

35. The Derivative
We can make our approximation of 𝑣 𝑡 0 exact by taking the limit as Δ𝑡→0
𝑣 𝑡 0 = lim Δ𝑡→0 𝑥 𝑡 0 +Δ𝑡 −𝑥 𝑡 0 Δ𝑡
We call this the “derivative of 𝑥 with respect to 𝑡”

36. The Derivative We denote the derivative as: 𝑣 𝑡 = d𝑥 d𝑡
d𝑥 and d𝑡 denote a “differential change”, which describes Δ𝑥 or Δ𝑡 in the limit where the difference goes to zero

37. The Derivative - Linearity
The derivative is a linear operation, this means:
d d𝑡 𝐴 𝑓 𝑡 =𝐴 d𝑓 d𝑡
d d𝑡 𝑓 𝑡 +𝑔(𝑡) = d𝑓 d𝑡 + d𝑔 d𝑡

38. The Derivative - Quadratic
Calculate 𝑣(𝑡) for:
𝑥 𝑡 =𝐴 𝑡 2

39. The Derivative - Polynomial
Calculate 𝑣(𝑡) for:
𝑥 𝑡 =𝐴 𝑡 𝑛

40. Power Rule
In general:
d d𝑡 𝑡 𝑛 =𝑛 𝑡 𝑛−1

41. Derivative of Sine & Cosine𝑣 𝑡
We know from graphical considerations that
𝑑 𝑑𝑡 sin 𝑡 looks like cos (𝑡) . How do we prove it?

42. Derivative of Sine & Cosine
In general:
d d𝑡 sin (𝑡) = cos (𝑡)
d d𝑡 cos (𝑡) = −sin (𝑡)

43. Second Derivative The second derivative of 𝑥(𝑡) is defined as:
d 2 𝑥 d 𝑡 2 = d d𝑡 d𝑥 d𝑡
We can relate the second derivative of 𝑥(𝑡) to other kinematic variables:
d 2 𝑥 d 𝑡 2 = d d𝑡 d𝑥 d𝑡 = d d𝑡 𝑣 𝑡 =𝑎(𝑡)

44. Third Derivative
The third derivative of position vs. time is called the jerk:
𝑗 𝑡 = d 3 𝑥 d 𝑡 3

45. The Chain Rule
Suppose we know height of the roller coaster as a function of its position 𝑦(𝑥). And we know 𝑥(𝑡).
How do we calculate 𝑑𝑦 𝑑𝑡 ?

46. The Chain Rule 𝑦 𝑥 𝑡

47. The Chain Rule
In general:
If we have 𝑦(𝑥) and 𝑥(𝑡),
𝑑𝑦 𝑑𝑡 = 𝑑𝑦 𝑑𝑥 𝑑𝑥 𝑑𝑡

48. The Chain Rule Consider: 𝑦 𝑥 =𝐴 𝑥 2 ; 𝑥 𝑡 = 1 2 𝑎 𝑡 2
𝑦 𝑥 =𝐴 𝑥 2 ; 𝑥 𝑡 = 1 2 𝑎 𝑡 2
Calculate d𝑦 d𝑡 using the chain rule.

49. The Chain Rule Consider: 𝑦 𝑡 =𝐴 b t 2 −c 2 What is 𝑥(𝑡) and 𝑦(𝑥)?
Calculate d𝑦 d𝑡

50. The Chain Rule Consider: 𝑦 𝑡 =𝐴 sin (𝑘 𝑡 2 ) What is 𝑥(𝑡) and 𝑦(𝑥)?
Calculate d𝑦 d𝑡

51. The Chain Rule 𝑦 𝑡 = 𝐴 sin 𝑡 2
Once you gain experience using the Chain Rule, you can skip steps.
Trick: Start from the outside and work your way in
Consider:
𝑦 𝑡 = 𝐴 sin 𝑡 2

52. 𝑦 𝑡 =sin(a t 2 ) The Chain Rule
Using this trick take the derivative of:
𝑦 𝑡 =sin(a t 2 )

53. The Chain Rule 𝑦 𝑡 = 1 𝑚 sin 5 𝑠 −1 𝑡
Numbers like (1 𝑚) and 5 𝑠 −1 are just like 𝐴 or 𝑎
𝑦 𝑡 = 1 𝑚 sin 5 𝑠 −1 𝑡
Calculate d𝑦 d𝑡 using the chain rule.

54. The Chain Rule
Given:
𝑥 𝑡 = 𝑎 𝑡 2 −𝑏 2
when is the object at rest?

55. Product Rule
How do we calculate the derivative of the product of two functions, 𝑓 𝑡 𝑔(𝑡)?
Apply the definition of the derivative!
𝑑 𝑑𝑡 𝑓 𝑡 𝑔 𝑡 = lim Δ𝑡→0 𝑓 𝑡+Δ𝑡 𝑔 𝑡+Δ𝑡 −𝑓 𝑡 𝑔 𝑡 Δ𝑡
Okay…now what do we do?

56. Product Rule 4 5
Recall that we can visualize the product of two numbers as the area of a rectangle.

57. Product Rule
4×5=20
Recall that we can visualize the product of two numbers as the area of a rectangle.

58. Product Rule 𝑓 𝑡 =𝑓 𝑡 𝑔(𝑡) 𝑔 𝑡
We can do the same thing with the product of two functions.

59. Product Rule
𝑓(𝑡+Δ𝑡)
𝑓(𝑡)
𝑔(𝑡)
𝑔(𝑡+Δ𝑡)

60. Product Rule 𝑓 𝑡+Δ𝑡 𝑓 𝑡 𝑔 𝑡 𝑔 𝑡+Δ𝑡 How do we geometrically picture:
𝑓 𝑡+Δ𝑡 𝑔 𝑡+Δ𝑡 −𝑓 𝑡 𝑔(𝑡)

61. Product Rule 𝑓 𝑡+Δ𝑡 𝑓 𝑡 𝑔 𝑡 𝑔 𝑡+Δ𝑡 Lets calculate:
𝑓 𝑡+Δ𝑡 𝑔 𝑡+Δ𝑡 −𝑓 𝑡 𝑔(𝑡)

62. Product Rule
𝑓 𝑡
𝑓 𝑡+Δ𝑡
𝑔 𝑡
𝑔 𝑡+Δ𝑡

63. The Product Rule
In general:
d d𝑡 𝑓 𝑡 𝑔 𝑡 = d𝑓 d𝑡 𝑔 𝑡 + d𝑔 d𝑡 𝑓(𝑡)

64. Product Rule goes to zero 𝑑𝑓 𝑑𝑡 𝑔(𝑡) in the limit: Δ𝑡→0 𝑓 𝑡 𝑑𝑔 𝑑𝑡 𝑓(𝑡)
𝑑𝑓 𝑑𝑡 𝑔(𝑡)
𝑓 𝑡
𝑑𝑔 𝑑𝑡 𝑓(𝑡)
𝑔 𝑡

65. The Product Rule Calculate the derivative of: ℎ 𝑡 =𝑎 𝑡 (𝑏 𝑡 2 +𝑐) 𝑓 𝑡
ℎ 𝑡 =𝑎 𝑡 (𝑏 𝑡 2 +𝑐)
𝑓 𝑡
𝑔 𝑡

66. The Product Rule
Calculate the derivative of:
ℎ 𝑡 = 𝑎 𝑡 3 𝑏 𝑡 2

67. The Product Rule
Calculate the derivative of:
ℎ 𝑡 =𝐴 𝑡 sin (𝜔 𝑡)

68. The Product Rule
Calculate the derivative of:
ℎ 𝑡 =𝑎 𝑡 𝑏 𝑡 3 +𝑐 2

69. Integration
An integral breaks something up into a bunch of tiny bits and then adds the contributions from all of the tiny bits.

70. Integration
What is the gravitational force exerted on a satellite orbiting a cube planet?
𝐹 = ?

71. Integration
What is the gravitational force exerted on a satellite orbiting a cube planet?

72. Integration
What is the gravitational force exerted on a satellite orbiting a cube planet?

73. Integration
What is the gravitational force exerted on a satellite orbiting a cube planet?

74. Integration
d d𝑡
𝑥(𝑡)
𝑣(𝑡)
∫𝑑𝑡

75. Integration For a constant velocity, we know that: 𝑣= Δ𝑥 Δ𝑡 Therefore,
Δ𝑥=𝑣 Δ𝑡

76. Integration
To calculate the displacement when velocity is not constant, we take a sum the tiny displacements made over many extremely small intervals 𝛿𝑡:
Δ𝑥=∑𝛿𝑥=∑𝑣 𝑡 𝛿𝑡
We call this process a Riemann Sum

77. Integration
In the limit that 𝛿𝑡→0, the Riemann Sum becomes a Riemann Integral
lim 𝑁→∞ 𝑛=0 𝑁 𝑣 𝑛 𝛿𝑡 𝛿𝑡 = 0 Δ𝑡 𝑣 𝑡 d𝑡
Where 𝛿𝑡= Δ𝑡 𝑁

78. Integration Notice that: 𝑡 0 𝑡 𝑓 𝑣 𝑡 d𝑡 =Δ𝑥=𝑥 𝑡 𝑓 −𝑥 𝑡 0
Or, equivalently:
𝑡 0 𝑡 𝑓 d𝑥 d𝑡 d𝑡 =𝑥 𝑡 𝑓 −𝑥 𝑡 0

79. Integration d𝑣 d𝑡 d𝑡 =𝑥 𝑡
Sometimes we ignore the end points of the integral:
d𝑣 d𝑡 d𝑡 =𝑥 𝑡

80. Integration In general: d𝑓 d𝑡 d𝑡 =𝑓(𝑡)
This is called the Fundamental Theorem of Calculus

81. Integration
What the Fundamental Theorem of Calculus is really saying is:
The integral is the inverse or “opposite” of the derivative
That is why the integral is sometime called the antiderivative

82. Integration
d d𝑡
𝑥(𝑡)
𝑣(𝑡)
∫𝑑𝑡

83. Integration
In practice, we use the Fundamental Theorem of Calculus to calculate integrals
If asked:
∫𝑣 𝑡 d𝑡= ?
Ask yourself: “What function has 𝑣(𝑡) as its derivative?”

84. Integration
Using the Fundamental Theorem of Calculus, derive a formula for:
𝑛 𝑡 𝑛−1 d𝑡
Ask yourself: “What function has the derivative 𝑛 𝑡 𝑛−1 ?”

85. Integration
Using the Fundamental Theorem of Calculus, derive a formula for:
𝑡 𝑛 d𝑡
Ask yourself: “What function has the derivative 𝑡 𝑛 ?”

86. Integration 𝑡 𝑛 𝑑𝑡 = 1 𝑛+1 𝑡 𝑛+1
You can always check your answer using the Fundamental Theorem of Calculus
𝑡 𝑛 𝑑𝑡 = 1 𝑛+1 𝑡 𝑛+1
To check, take the derivative of 1 𝑛+1 𝑡 𝑛+1

87. Integration
What if I asked:
𝑡 0 𝑡 𝑓 𝑡 𝑛 d𝑡 =

88. Integration - Example
What is:
𝑎 𝑡 2 𝑑𝑡=

89. Integration
What about
𝐴 cos 𝑡 d𝑡

90. Integration
What about
𝐴 sin 𝑡 d𝑡

91. Integration For Sine & Cosine, we have cos 𝑡 d𝑡 =sin⁡(𝑡)
sin 𝑡 d𝑡 =−cos⁡(𝑡)

92. Integration - Example
Suppose an object is moving with a velocity given by:
𝑣 𝑡 = 5 m s 𝑡 3
How far does the object travel between 𝑡 0 =0 𝑠 and 𝑡 𝑓 =3 𝑠

93. Integration - Example
Δ𝑥= 𝑡 0 =0 𝑠 𝑡 𝑓 =3 𝑠 𝑣 𝑡 d𝑡 = 𝑡 0 =0 𝑠 𝑡 𝑓 =3 𝑠 5 m s 𝑡 3 d𝑡 = 5 m s 𝑡 0 =0 𝑠 𝑡 𝑓 =3 𝑠 𝑡 3 d𝑡

94. Integration - Example
Δ𝑥= 5 m s 𝑡 𝑓 4 − 𝑡 = m s 4 (81 s 4 −0 s 4 ) = m

95. Graphical Analysis of Integration
Just like the derivative is graphically linked to the steepness of a graph, the integral also has a graphical interpretation.
Consider an object traveling at the constant velocity. The displacement is given by:
Δ𝑥=𝑣 Δ𝑡

96. Graphical Analysis of Integration
Recall that the product of two values can be visualized as the area of a rectangle. 𝑣
Area=𝑣 Δ𝑡 =Δ𝑥 Δ𝑡

97. Graphical Analysis of Integration
When given a graph of 𝑣 vs. 𝑡, we can visualize Δ𝑥 as the area under 𝑣(𝑡) over a time interval, Δ𝑡 𝑣 𝑣 𝑡 Δ𝑡

98. Graphical Analysis of Integration
Recall that integration is defined in terms of the Riemann Sum:
0 Δ𝑡 𝑣 𝑡 d𝑡 = lim 𝑁→∞ 𝑛=0 𝑁 𝑣 𝑛 𝛿𝑡 𝛿𝑡

99. Graphical Analysis of Integration
0 Δ𝑡 𝑣 𝑡 d𝑡 = lim 𝑁→∞ 𝑛=0 𝑁 𝑣 𝑛 𝛿𝑡 𝛿𝑡
Each 𝑣 𝑛 𝛿𝑡 𝛿𝑡 in the sum can be visualized as the area of a rectangle
Therefore, the Riemann sum can be visualized as the sum of a series of rectangular areas

100. Graphical Analysis of Integration
Velocity
Time

101. Graphical Analysis of Integration
Velocity
Time

102. Graphical Analysis of Integration
Velocity
Time

103. Graphical Analysis of Integration
In general:
𝑡 0 𝑡 𝑓 𝑣 𝑡 d𝑡
Is equal to the area under 𝑣(𝑡) between 𝑡 0 and 𝑡 𝑓 .

104. 𝑣 𝑡 Δ𝑥 𝑡

105. Graphical Analysis of Integration
When considering graphs of 𝑥(𝑡) and 𝑣 𝑡 remember:
𝑥(𝑡) is the area under 𝑣(𝑡)
𝑣(𝑡) is the slope of 𝑥(𝑡)

106. Graphical Analysis of Integration𝑣 𝑡 Δ𝑥 𝑡

107. Graphical Analysis of Integration
An object returns to its initial position when the area above the 𝑡 – axis is equal to the area below the 𝑡 – axis.