Chapter 1 Introduction

If you are new to studying physics, you will need to be familiar with some basic algebra. In this chapter, we will review important math techniques. Don't let numbers and calculations intimidate you. This chapter will help you become more familiar with math.


1. Units
2. Scientific Notation
3. Significant Digits
4. Graph
5. Trigonometry
6. Chapter 1 Quiz

  Section Section 1: Units

The metric system of measurement is the standard in the world. The fundamental units include the second (s) for time, the meter (m) for length, and the kilogram (kg) for mass.

You should know how to convert from one unit to another.



3600 seconds = 60 minutes = 1 hour
100 centimeters = 1 meter    
1000 grams = 1 kilogram    


  Section Section 2: Scientific Notation

When expressing an extreme large number such as the mass of Earth, or a very small number such as the mass of an electron, scientists use the scientific notation. The basic format of scientific notation is M * 10n, where M is any real numbers between 1 and 10 and n is a whole number.



100 = 1
101 = 10
102 = 10 * 10 = 100
103 = 10 * 10 * 10 = 1000
10-1 = 1 / 10 = 0.1
10-2 = 1 / 10 / 10 = 0.01
10-3 = 1 / 10 / 10 / 10 = 0.001

For example, the mass of Earth is about

6,000,000,000,000,000,000,000,000 kg

and can be written as 6.0 * 1024 kg.

Also, the mass of an electron is

0.000000000000000000000000000000911 kg

and can be expressed as 9.11 * 10-31 kg.

QUESTION: Express 8.213 * 102 in decimal number.

QUESTION: Solve 4 * 102 + 3.2 * 103.


  Section Section 3: Significant Digits

The significant digits represent the valid digits of a number. The following rules summarize the significant digits:



  1. Nonzero digits are always significant.
  2. All final zeros after the decimal points are significant.
  3. Zeros between two other significant digits are always significant.
  4. Zeros used solely for spacing the decimal point are not significant.

The table below is an example:

values # of significant digits
5.6 2
0.012 2
0.0012003 5
0.0120 3
0.0012 2
5.60 3

In addition and subtraction, round up your answer to the least precise measurement. For example:

24.686 + 2.343 + 3.21 = 30.239 = 30.24

because 3.21 is the least precise measurement.

In multiplication and division, round it up to the least number of significant digits. For example:

3.22 * 2.1 = 6.762 = 6.8

because 2.1 contains 2 significant digits.

In a problem with the mixture of addition, subtraction, multiplication or division, round up your answer at the end, not in the middle of your calculation. For example:

3.6 * 0.3 + 2.1 = 1.08 + 2.1 = 3.18 = 3.2.

QUESTION: Solve 5.123 + 2 + 0.00345 - 3.14.

QUESTION: Solve -9.300 + 2.4 * 3.21.



Section 4: Graph

Three types of mathematical relationships are most common in physics.

One of them is a linear relationship, which can be expressed by the equation y = mx + b where m is the slope and b is the y-intercept.

A graph representing the Linear Relationship

Another relationship is the quadratic relationship. The equation is y = kx2, where k is a constant.

A graph representing the Quadratic Relationship

The third equation is an inverse relationship, expressed by xy = k, where k is a constant.

A graph representing the Inverse Relationship


  Section Section 5: Trigonometry

Trigonometry is also important in physics. When you have a right-angled triangle, the following relationships are true:

Formula for sine A right triangle to explain Sine, Cosine, and Tangent
Formula for Cosine
Formula for Tangent

Trigonometry will become important when you study vectors.

QUESTION: You are looking up at the top of a tree that is 10 m apart from you. If the tree is 15 m taller than you, at what angle are you looking upward? (e.g. 30.0)

Chapter 1 Quiz:

1. Convert the following scientific notations into decimal numbers.

a. 5.8 * 102
b. 1.2 * 10-3
c. 9.30001 * 100

2. State the number of significant digits:

a. 1.10
b. 0.000000023000
c. 12004

3. Solve the following problem with correct numbers of significant digits.

a. 7.29 + 2.0001 - 3.2
b. 100.3 - 5.2 * 3.11
c. 2.1 * 5.2 - 1.45 / 0.02303

4. Solve the following equation for x.

a. 5 = 2x - 1
b. 36 = 4x2
c. 5x = 15

5. You are looking up at the top of a building at 30 degrees. If the building is 20 m tall, how far are you from the building? Assume that you are 1.70 m tall. (e.g. "10.5 m")


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Chapter 2 Velocity

By Keiji Oenoki []

We will begin our study of physics with a study of motion.

Look around you, and you will find that almost everything is in motion: flying birds, running people, falling books, etc. We will analyze their motions and think about how fast the object moves and how far.

1. Distance and Displacement
2. Average Velocity and Instantaneous Velocity
3. Position-time Graph
4. Velocity-time Graph
5. Relative Motion
6. Chapter 2 Quiz


Section 1. Distance and Displacement

Distance and displacement are different. When you traveled 50 km to the East and then 20 km to the West, the total distance you traveled is 70 km, but your displacement is 30 km East.

A picture explaining distance and displacement

In physics, we say that distance is a scalar and displacement is a vector. Scalar has a magnitude and vector has both a magnitude and a direction. Scalar is one dimensional and vector is two dimensional.


QUESTION: A car moved 50 km to the North. What is its displacement?
(e.g. "10 km East") QUESTION: A car moved 20 km East and 70 km West. What is the distance?

Section 2. Average Velocity and Instantaneous Velocity

Velocity shows how fast an object is moving to which direction. Average velocity can be calculated by dividing displacement over time.

Formula for Velocity

For example, when a car moved 50 km in 2 hours, the average velocity is 25.5 km/h because 50 km divided by 2 h

The instantaneous velocity shows the velocity of an object at one point. For example, when you are driving a car and its speedometer swings to 90 km/h, then the instantaneous velocity of the car is 90 km/h.

QUESTION: A car moved 20 km East and 60 km West in 2 hours. What is its average velocity?   km/h QUESTION:

How far will a car travel in 15 min at 20 m/s?    km


Section 3. Position-time Graph

A position-time graph simply shows the relationship between time and position. From the following data, for example,

time (s) 0 1 2 3 4 5
position (s) 0 20 50 130 150 200

You can draw the following graph:

A position-time graph

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For math-crazy only: The tangent of a position-time graph represents velocity since

Formula to get velocity from a position-time graph

QUESTION: Answer the questions using the previous graph:

What is the average velocity during the first 2 seconds?    m/s

What is the average velocity of the whole trip?    m/s

Section 4. Velocity-time Graph

A velocity-time graph shows the relationship between velocity and time. For example, if a car moves at constant velocity of 5 m/s for 10 seconds, you can draw a velocity-time graph that looks like this:

A velocity-time graph

The area below the line represents the displacement the object traveled since it can be calculated by xy, or (time * velocity) which equals to displacement.

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Section 5. Relative Motion

When the car A is at 50 km/h and the car B is at 30 km/h at opposite direction, the velocity of the car A relative to the car B is 80 km/h.

QUESTION: If you are walking at constant velocity of 5 km/h and a car passed you by at the speed of 20 km/h from behind, what is the car's velocity from your viewpoint?  km/h

QUESTION: If you are running at constant velocity of 5 m/s, what is your relative velocity to Earth?  m/s


Chapter 2 Quiz:

1. If you move 10 km North, 10 km East, and 10 km South,

a. What is your displacement? (e.g. "10 km South")
b. What is the total distance you traveled? km

2. The below graph represents the position of a mouse at a given time.

A position-time graph of the mouse

a. What is the average velocity between t = 0 to t = 4? m/s
b. What is the distance you traveled from t = 3 to t = 7? m
c. What is the total displacement? m

3. Answer the questions using the following graph.

A velocity-time graph

a. What is the distance it traveled between t = 0 to t = 2? m
b. What is the instantaneous velocity when t = 4? m/s

4. Jack and Michael are both running together at constant velocity of 10 m/s.

a. What is Jack's relative velocity to Michael? m/s
b. What is Michael's relative velocity to Jack? m/s

5. A rat and a cat are 25 m apart. When the rat started to run at 2 km/h, the cat started to chase him at 10 km/h. Can the cat catch the rat in 10 seconds?

("Yes" or "No")

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