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 [ ]

Rules of Dimensionality

1. Variables on opposite sides of an equals sign must have the same dimensionality

2. Variables on opposite sides of a + or - must have the same dimensionality

Rules of Dimensionality

Lets check the formula:

Rules of Dimensionality

3. Pure number () are always dimensionless

4. Special functions (sine, cosine, exponential, etc.) are always dimensionless

5. The argument of special functions are always dimensionless

Dimensional Analysis

We can use the rules of dimensionality to find the dimensions of an unknown quantitiy in a formula:

Dimensional Analysis

Dimensional Analysis

Dimensional Analysis

Consider a mass swinging on the end of a stringThe period is the amount of timetakes for the mass to complete onefull oscillationWhat variables do yoususpect the period of the motion will depend on?

Dimensional AnalysisIn general we may assume:

Using dimensional considerations, we can solve for and

Position, Velocity, & Acceleration

• In Physics it is important to be able to relate position, velocity, & acceleration

• A mathematical description of this relationship requires the use of calculus

• In this section we will discuss the graphical relationship between a position vs. time graph and a velocity vs. time graph

Graphical Analysis

Δ 𝑦

Δ𝑥

• Recall that:

𝑦

𝑥

Graphical Analysis

• For a position vs. time graph:

• For an velocity vs. time graph:

Graphical Analysis

𝑡

𝑥

Δ 𝑡

Δ𝑥

𝑣𝑎𝑣𝑔=Δ𝑥Δ𝑡

Graphical Analysis

𝑡

𝑥

Δ 𝑡1

Δ𝑥1

𝑣𝑎𝑣𝑔 ,1<𝑣𝑎𝑣𝑔 ,2

Δ𝑥2

Δ 𝑡2

Graphical Analysis

𝑡

𝑥

𝑡

𝑠𝑙𝑜𝑝𝑒=𝑣 (𝑡)

Graphical Analysis

• is the slope of the tangent line at • is graphically understood as the steepness of the

vs graph.

Graphical Analysis

𝑥

𝑡

What does look like?

Graphical Analysis

Identify where positive, negative, & zero

𝑡

𝑥

Graphical Analysis

Sketch a graph of 𝑡

𝑣

𝑡

The Derivative

• We can approximate as the average velocity over a time an interval starting at

The Derivative

𝑡

𝑥

𝑡 0

The Derivative

𝑡

𝑥

Δ 𝑡

The Derivative

𝑡

𝑥

Δ 𝑡

The Derivative

𝑡

𝑥

Δ 𝑡

The Derivative

𝑡

𝑥

Δ 𝑡

The Derivative

𝑡

𝑥

Δ 𝑡

The Derivative

• We can make our approximation of exact by taking the limit as

We call this the “derivative of with respect to ”

The Derivative

• We denote the derivative as:

• and denote a “differential change”, which describes or in the limit where the difference goes to zero

The Derivative - Linearity

The derivative is a linear operation, this means:

The Derivative - Quadratic

Calculate for:

The Derivative - Polynomial

Calculate for:

Power Rule

In general:

Derivative of Sine & Cosine

𝑣

𝑡

We know from graphical considerations that looks like . How do we prove it?

Derivative of Sine & Cosine

In general:

Second Derivative

The second derivative of is defined as:

We can relate the second derivative of to other kinematic variables:

Third Derivative

The third derivative of position vs. time is called the jerk:

The Chain Rule

Suppose we know height of the roller coaster as a function of its position . And we know .How do we calculate ?

The Chain Rule

𝑦

𝑥

𝑥

𝑡

The Chain Rule

In general:If we have and ,

The Chain Rule

Consider:

Calculate using the chain rule.

The Chain Rule

Consider:

What is and ?

Calculate

The Chain Rule

Consider:

What is and ?

Calculate

The Chain Rule

Once you gain experience using the Chain Rule, you can skip writing down and .

The trick: work from the outside

Consider:

The Chain Rule

Consider:

Calculate using the chain rule.

The Chain Rule

Calculate the derivative of:

The Chain Rule

Consider:

Determine when

How do we calculate the derivative of the product of two functions, ?

Apply the definition of the derivative!

Okay…now what do we do?

Product Rule

Recall that we can visualize the product of two numbers as the area of a rectangle.

4

Product Rule

5

Recall that we can visualize the product of two numbers as the area of a rectangle.

Product Rule

1234567 891011121314 151617181920

4×5=20

We can do the same thing with the product of two functions.

Product Rule

¿ 𝑓 (𝑡 )𝑔 (𝑡)𝑓 (𝑡 )

𝑔 (𝑡 )

Consider two functions & which are both increasing.

Product Rule𝑓 (𝑡)

𝑡

𝑔 (𝑡)

𝑡

Product Rule

𝑓 (𝑡)𝑓 (𝑡+Δ𝑡 )

𝑔 (𝑡)𝑔 (𝑡+Δ𝑡)

Product Rule

𝑓 (𝑡 )

𝑔 (𝑡 )

𝑓 (𝑡+Δ𝑡 )

𝑔 (𝑡+Δ 𝑡 )

How do we geometrically picture:

Product Rule

𝑓 (𝑡 )

𝑔 (𝑡 )

𝑓 (𝑡+Δ𝑡 )

𝑔 (𝑡+Δ 𝑡 )

Lets calculate:

The Product Rule

In general:

Product Rule𝑑𝑓𝑑𝑡

𝑔 (𝑡)

𝑓 (𝑡 )

𝑔 (𝑡 )

𝑑𝑔𝑑𝑡

𝑓 (𝑡)

goes to zeroin the limit:

The Product Rule

Calculate the derivative of:

𝑓 (𝑡 ) 𝑔 (𝑡 )

The Product Rule

Calculate the derivative of:

The Product Rule

Calculate the derivative of:

The Product Rule

Calculate the derivative of: