AP Physics C

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

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

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

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

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

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

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

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

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

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

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?

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

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

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

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

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

Graphical Analysis 𝑥 𝑣 𝑎𝑣𝑔 = Δ𝑥 Δ𝑡 Δ𝑥 𝑡 Δ𝑡

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

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

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.

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

𝑥 𝑡 𝑣 𝑡

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

𝑥 𝑡 𝑣 𝑡

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

𝑣 𝑡 𝑣 𝑡

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

The Derivative 𝑥 𝑡 𝑡 0

The Derivative 𝑥 𝑡 Δ𝑡

The Derivative 𝑥 𝑡 Δ𝑡

The Derivative 𝑥 𝑡 Δ𝑡

The Derivative 𝑥 𝑡 Δ𝑡

The Derivative 𝑥 𝑡 Δ𝑡

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”

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

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

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

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

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

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

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

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)

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

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 ?

The Chain Rule 𝑦 𝑥 𝑡

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

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.

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

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

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

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

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.

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

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?

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

Product Rule 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 4×5=20 Recall that we can visualize the product of two numbers as the area of a rectangle.

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

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

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)

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

Product Rule 𝑓 𝑡 𝑓 𝑡+Δ𝑡 𝑔 𝑡 𝑔 𝑡+Δ𝑡

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

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

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

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

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

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

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

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

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

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

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

Integration d d𝑡 𝑥(𝑡) 𝑣(𝑡) ∫𝑑𝑡

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

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

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

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

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

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

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

Integration d d𝑡 𝑥(𝑡) 𝑣(𝑡) ∫𝑑𝑡

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?”

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 ?”

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

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

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

Integration - Example What is: 𝑎 𝑡 2 𝑑𝑡=

Integration What about 𝐴 cos 𝑡 d𝑡

Integration What about 𝐴 sin 𝑡 d𝑡

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

Integration - Example Suppose an object is moving with a velocity given by: 𝑣 𝑡 = 5 m s 4 𝑡 3 How far does the object travel between 𝑡 0 =0 𝑠 and 𝑡 𝑓 =3 𝑠

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 4 𝑡 3 d𝑡 = 5 m s 4 𝑡 0 =0 𝑠 𝑡 𝑓 =3 𝑠 𝑡 3 d𝑡

Integration - Example Δ𝑥= 5 m s 4 1 4 𝑡 𝑓 4 − 𝑡 0 4 = 5 4 m s 4 (81 s 4 −0 s 4 ) =101.25 m

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

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

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

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

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

Graphical Analysis of Integration Velocity Time

Graphical Analysis of Integration Velocity Time

Graphical Analysis of Integration Velocity Time

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 .

𝑣 𝑡 Δ𝑥 𝑡

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)

Graphical Analysis of Integration 𝑣 𝑡 Δ𝑥 𝑡

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.