# Unit 1.1 | Understanding vectors and the standard units used in Physics

###### Jason Kuma

Writer | Coach | Builder | Fremont, CA

###### Article Content
In this beginner’s guide, we’ll delve into the fascinating intricacies of movement. From a tiny ant marching along the ground to a rocket soaring into space, everything is in motion!

By the end of this unit you will be able to solve complex kinematic problems using equations and graphs.

In this first lesson out of five, I’ll help you build a robust understanding of vectors in the simplest way possible.

### Unit 1 Breakdown

You are on Lesson 1 of 5

#### In this lesson you will learn

• Kinematics – what it is?
• Vectors – what they are and how/why we use them in Physics
• Vector manipulation – adding and splitting up vectors
• Units – motion graphs and equations

### Introduction

Hello, young physicists! Welcome to the exciting world of physics. We will embark on a journey to understand the universe and the principles that govern it.

Kinematics, or the study of motion, is the first part of this journey.

It makes up 12-18% of the AP Physics 1 content. Or around 10% of a regular Physics curriculum.

In order to understand Kinematics (and the rest of Physics) we must first cover 2 important topics: Vectors and Standard Units.

### What are Vectors?

Vectors are simply arrows! They will be used EVERYWHERE in physics, not just kinematics.

Vectors have two things: a length (also called a magnitude or scalar) and a direction.

This is how to write a vector: [katex] \vec{A}[/katex].

This is how to write a scalar: A OR [katex] \lvert \vec{A} \rvert[/katex].

#### Length and Direction

The length can represent anything like distance, speed, forces, and more.

And the direction can be given as any of the 4 things below:

• a general direction (up, down, left, right)
• a compass direction (north, south, east, west)
• an angle (30° below the horizon)
• a simple positive or negative sign (+/-)

Vector example: riding on a bike at 5 m/s north. Notice how BOTH the magnitude (5 miles) AND the direction (north) is specified.

### Why Vectors

You might be wondering where and why we use vectors in physics.

Using vectors allows us to view motion in multiple directions.

Look at the image below. The soccer ball is kicked at a velocity [katex]v_0[/katex] that is 30º above the soccer field.

Note that velocity means speed and [katex]v_0[/katex] is the variable for “initial velocity.” It is a placeholder for the numerical value of the velocity of the soccer ball.

Physicists can take the [katex]v_0[/katex] velocity vector and separate it into two parts: the [katex]H[/katex] vector and the [katex]R[/katex] vector as shown in the image above. These are what we call vector components.

#### Vector Components

Since we can break 1 vector down into 2 components of this motion, we can see that the ball is really moving both up and to the right at the same time.

We can use these components (or parts of the velocity vector) to solve real world problems, like how far did the ball travel before landing on the ground?

### LRN Vector Math

In Physics, there are 4 things you will be doing with vectors. The most important one is the last one and there is a short video below showing how to do it.

1. [katex] number \times \vec {A} [/katex] : Multiply a vector by a scalar to increase the size the of the vector.
2. [katex] \vec{A} + \vec{B} [/katex] : Add two vectors along the same axis to get a resultant vector.
3. [katex] \vec{A} \times \vec{B} [/katex] : Add two perpendicular vectors to get a resultant vector.
4. [katex] \vec{A} = \vec{A_x} \times \vec{A_y}[/katex] : Take 1 vector and break it down into the horizontal and vertical components

Okay now that you what we can do with vectors, lets put into practice! Watch the short video below, and try the example problems after.

### PQ Vector Components

After watching the video above you should have a solid understanding of adding vectors and breaking them into components. Attempt all the practice questions below and be sure to understand all missed ones before moving on.

(1) Find the resultant vector (magnitude AND direction) given the following components:

1. Ax = 5.7, Ay = 3.2; Answer: (A = 6.6 at 30.8 degrees above the +x axis)
2. Bx = -10, By = -3; Answer: (B = 10.4 at 16.7 degrees below the -x axis)
3. Cx = 11.65, Cy = -20.17; Answer: (C = 23.3 at 60 degrees below the +x axis)

(2) Find the X and Y components of the following vectors:

1. 35 m/s at 57° from the x-axis; Answer: (X: 19.1 m/s, Y: 29.4 m/s)
2. 12 m/s at 34° S of W; Answer: (X: -10 m/s, Y: -6.7 m/s)
3. 20 m/s 275° from the x-axis; Answer: (X: 1.75 m/s, Y: -20 m/s)
4. A plane takes off with a velocity, v, of 40 m/s at an angle of 15° above the ground; Answer: (vx = 38.6 m/s, vy: 10.4 m/s)

(3) Challenge: Listed below are 4 distance vectors.

• A: 15 m at 60° above the +x-axis
• B: 10 m east
• C: 25 m south
• D: 50 meters at 28° South of West

Using the vectors above, preform the following operations:

1. A + B + C + D; Answer: (29 m at 23° above the -x-axis or 23° N of W)
2. A – C; Answer: (38.7 m at 78.8° above the +x-axis or 78° N of E)
3. D + 2C; Answer: (51.4 m at 31° below the -x-axis or 31° S of W)

Hint: Draw out each vector on a graph. Break each vector into x and y components, then preform the operations in each direction.

### Using the Standard Units of Measurement

If you’re from the United States you may be used to seeing miles, miles per hour, feet, pounds etc.

This is the imperial system of measurement. And it makes calculations and conversions difficult.

Instead, in physics, we use the metric system (kilometers, meters, kilograms, etc) that the rest of the world is familiar with.

#### Why Metric?

The metric system is much easier to convert between orders of magnitude.

For example, how many centimeters are in a meter? Well we can simply look at the prefix “centi” and tell that there are 100 centimeters in 1 meter.

So in Physics, you’ll commonly see the following standard units

• Distance is measured in meters (m)
• Speed is measured in meters per second (m/s)
• Weight is measured in kilograms (kg)
• Time is measured in seconds (s)

From now on, when solving a problem, make sure to convert all give numbers to the standard units above!

### Unit Conversions

As mentioned earlier, converting units is pretty easy in the metic system.

You just have to memorize what each prefix means. The chart below will help with that.

As you do more Physics problems, unit conversions will become super easy!

AND ff you are having trouble converting units, watch this simple video explanation

### PQ Converting Units

1. Convert 320 grams to kg
2. Convert 145 mm to meters
3. Convert 100 km/hr the standard unit of velocity used in Physics
4. Convert 50 cm/min to m/s
5. Convert 365 days to seconds
6. Given that 1 inch = 2.54 cm convert 4674 inches to meters
7. The volume of a steel cube is 300 cm3. Convert this to m3
8. The density of a iron sphere is 7.9 g/cm3. Convert this to kg/m3

1. .32 kg
2. .144 m
3. 27.8 m/s (In Physics we use ‘ m/s ‘ as a unit of speed and velocity)
4. .0083 m/s

5. 3.15 x 107 s or 31500000 s
6. 118.7 m
7. 3 x 10-4 m3 or .0003 m3
8. 7900 kg/m3

### Lesson 1.2 Preview

In the next lesson we will begin to cover kinematics by exploring the 3 kinematic variables: displacement, velocity, and acceleration. We will use standard units. In lesson 1.4 we will begin to utilize our knowledge of vectors to real world problems.

###### Jason Kuma

Writer | Coach | Builder | Fremont, CA

## Units in AP Physics 1

Unit 1 – Linear Kinematics

Unit 2 – Linear Forces

Unit 3 – Circular Motion

Unit 4 – Energy

Unit 5 – Momentum

Unit 6 – Torque

Unit 7 – Oscillations

Unit 8 – Fluids

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KinematicsForces
[katex]\Delta x = v_i t + \frac{1}{2} at^2[/katex][katex]F = ma[/katex]
[katex]v = v_i + at[/katex][katex]F_g = \frac{G m_1m_2}{r^2}[/katex]
[katex]a = \frac{\Delta v}{\Delta t}[/katex][katex]f = \mu N[/katex]
[katex]R = \frac{v_i^2 \sin(2\theta)}{g}[/katex]
Circular MotionEnergy
[katex]F_c = \frac{mv^2}{r}[/katex][katex]KE = \frac{1}{2} mv^2[/katex]
[katex]a_c = \frac{v^2}{r}[/katex][katex]PE = mgh[/katex]
[katex]KE_i + PE_i = KE_f + PE_f[/katex]
MomentumTorque and Rotations
[katex]p = m v[/katex][katex]\tau = r \cdot F \cdot \sin(\theta)[/katex]
[katex]J = \Delta p[/katex][katex]I = \sum mr^2[/katex]
[katex]p_i = p_f[/katex][katex]L = I \cdot \omega[/katex]
Simple Harmonic Motion
[katex]F = -k x[/katex]
[katex]T = 2\pi \sqrt{\frac{l}{g}}[/katex]
[katex]T = 2\pi \sqrt{\frac{m}{k}}[/katex]
ConstantDescription
[katex]g[/katex]Acceleration due to gravity, typically [katex]9.8 , \text{m/s}^2[/katex] on Earth’s surface
[katex]G[/katex]Universal Gravitational Constant, [katex]6.674 \times 10^{-11} , \text{N} \cdot \text{m}^2/\text{kg}^2[/katex]
[katex]\mu_k[/katex] and [katex]\mu_s[/katex]Coefficients of kinetic ([katex]\mu_k[/katex]) and static ([katex]\mu_s[/katex]) friction, dimensionless. Static friction ([katex]\mu_s[/katex]) is usually greater than kinetic friction ([katex]\mu_k[/katex]) as it resists the start of motion.
[katex]k[/katex]Spring constant, in [katex]\text{N/m}[/katex]
[katex] M_E = 5.972 \times 10^{24} , \text{kg} [/katex]Mass of the Earth
[katex] M_M = 7.348 \times 10^{22} , \text{kg} [/katex]Mass of the Moon
[katex] M_M = 1.989 \times 10^{30} , \text{kg} [/katex]Mass of the Sun
VariableSI Unit
[katex]s[/katex] (Displacement)[katex]\text{meters (m)}[/katex]
[katex]v[/katex] (Velocity)[katex]\text{meters per second (m/s)}[/katex]
[katex]a[/katex] (Acceleration)[katex]\text{meters per second squared (m/s}^2\text{)}[/katex]
[katex]t[/katex] (Time)[katex]\text{seconds (s)}[/katex]
[katex]m[/katex] (Mass)[katex]\text{kilograms (kg)}[/katex]
VariableDerived SI Unit
[katex]F[/katex] (Force)[katex]\text{newtons (N)}[/katex]
[katex]E[/katex], [katex]PE[/katex], [katex]KE[/katex] (Energy, Potential Energy, Kinetic Energy)[katex]\text{joules (J)}[/katex]
[katex]P[/katex] (Power)[katex]\text{watts (W)}[/katex]
[katex]p[/katex] (Momentum)[katex]\text{kilogram meters per second (kgm/s)}[/katex]
[katex]\tau[/katex] (Torque)[katex]\text{newton meters (Nm)}[/katex]
[katex]I[/katex] (Moment of Inertia)[katex]\text{kilogram meter squared (kgm}^2\text{)}[/katex]
[katex]f[/katex] (Frequency)[katex]\text{hertz (Hz)}[/katex]

General Metric Conversion Chart

Example of using unit analysis: Convert 5 kilometers to millimeters.

1. Start with the given measurement: [katex]\text{5 km}[/katex]

2. Use the conversion factors for kilometers to meters and meters to millimeters: [katex]\text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}}[/katex]

3. Perform the multiplication: [katex]\text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}} = 5 \times 10^3 \times 10^3 \, \text{mm}[/katex]

4. Simplify to get the final answer: [katex]\boxed{5 \times 10^6 \, \text{mm}}[/katex]

Prefix

Symbol

Power of Ten

Equivalent

Pico-

p

[katex]10^{-12}[/katex]

Nano-

n

[katex]10^{-9}[/katex]

Micro-

µ

[katex]10^{-6}[/katex]

Milli-

m

[katex]10^{-3}[/katex]

Centi-

c

[katex]10^{-2}[/katex]

Deci-

d

[katex]10^{-1}[/katex]

(Base unit)

[katex]10^{0}[/katex]

Deca- or Deka-

da

[katex]10^{1}[/katex]

Hecto-

h

[katex]10^{2}[/katex]

Kilo-

k

[katex]10^{3}[/katex]

Mega-

M

[katex]10^{6}[/katex]

Giga-

G

[katex]10^{9}[/katex]

Tera-

T

[katex]10^{12}[/katex]

1. Some answers may be slightly off by 1% depending on rounding, etc.
2. Answers will use different values of gravity. Some answers use 9.81 m/s2, and other 10 m/s2 for calculations.
3. Variables are sometimes written differently from class to class. For example, sometime initial velocity [katex] v_i [/katex] is written as [katex] u [/katex]; sometimes [katex] \Delta x [/katex] is written as [katex] s [/katex].
4. Bookmark questions that you can’t solve so you can come back to them later.
5. Always get help if you can’t figure out a problem. The sooner you can get it cleared up the better chances of you not getting it wrong on a test!

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