Vector

Scalar Quantities are quantities which represent only magnitude, such as kilograms and metres. This way of presenting quantities only tells you how much of something has happened, it does not tell you in which direction it happened. To solve problems which involve a direction, we use:

Vector Quantities. These are simply distance, speed and acceleration quantities which have a specific direction. Here are some typical vector quantities: 7m [east], 56km/h [north], 4m/s [E56°N].

Concepts and Notation
Lets learn about some useful vector concepts:

* Position: This is the location of an object relative to a reference point. An example of using position would be saying "My cat is located 2m [east] of me".
* Displacement: Closely tied to position. Displacement is the "space between positions", if you are situated 7m [north] of your house ( position 1 ), you walked for a while and you stopped 17m [north] or your house ( position 2 ) then your displacement is 10m [north]. Displacement is different from distance, this will be discussed later.
* Velocity: Simply displacement over time. How much has your position changed over time. Here are some velocities: 44km/h [south], 14m/s [southeast]. It differs from speed.
* Acceleration: velocity over time. What sets it apart from scalar acceleration is that acceleration will occur even when the magnitude of the velocity is identical - when the direction of the velocity changes.

Notation:
A vector variable will always have a line/little arrow on top of its variable: v I use the above variable to represent velocity. The vector quantity itself will have its direction written in square brackets after the magnitude: 75m [east], 33km/h [west]. When the direction does not directly correspond to one of the four directions (NSEW) we can use angles to describe the direction. The notation that we use is:

[ direction1 angle from direction1 to direction2 direction2 ]

Lets analyze a vector quantity, 23m [S55°E].
23m is the displacement, this is the scalar (directionless) portion of the vector quantity. [S55°E] is the direction portion, which turns this into a vector quantity. The "S" stands for south, 55 is the angle which the direction takes from south to "E", east. The actual direction is 55 degrees east of the south direction. The above displacement can also be written as 23m [E35°S] ( we go from east 35 degrees to the south ).
Differences Between Vectors and Scalars
You already know the most important difference between vectors and scalars - vectors have a direction and scalars don't. However there are other important distinctions.

Displacement is very different than distance; distance is the actual amount of travel that has happened during motion, displacement is how far you are from your reference point. So that if you start off in your home, go around and return you have passed a certain distance, but your displacement is zero because you have returned to your starting point. No matter how you got to a certain point, displacement will always be the most direct distance to that point ( from where you started ). Keep in mind that displacement will also have an angle associated with it.

Velocity is not speed either. Velocity is the change in displacement, so that if you went in zigzags and spent an hour getting to a location 1km away from your starting point then your velocity will be 1km/1hour [direction]. The velocity takes into account only the final position and the total time it got to get there.
Using Vectors
You should use vectors any time you use angles and directions of motion, whenever your motion is not forward/backward in the same direction. Vectors are highly useful when they are drawn out, then you can use trigonometric ratios to find out resultant vectors.
What do vectors look like?
When you draw vectors ( which are just a way to describe motion in a visual way ) they look like a line with a "head" and a "tail". The line begins in a tail, which shows where you started from and ends in a head ( which is an arrow ) that shows where the motion ended.
A vector looks like a line with an arrow at the end
The little d with an arrow on top shows that this vector is a displacement vector, it looks like its in the direction [Right 20° Up] ( we are going 20 degrees Up from the Right direction ). The length of the line is ideally the magnitude of the vector ( the scalar part of it ), so when drawing vectors you should make them reasonably proportional ( e.g. a 3m vector should be smaller than a 7m one ).

An important mathematical tool when using vectors is the absolute value notation. An absolute value of a vector is written like so: |sheeshmaria!| and it is a way to represent just the magnitude of the vector, without direction. We will be taking the absolute value in an example question.
Vector addition
This is the actual application of vectors, adding and subtracting vector quantities. When you have two vectors to add you arrange them head to tail and you draw a line from the first tail to the last head:
When adding vectors arrange them head to tail, draw a new vector from the first tail to the last head
The vectors, again, represent displacement. The vectors represent the motion of a person: at first the person moved roughly to the east ( diplacement 1, d1 ), then the person moved northeast (d2) and stopped. Now, to figure out where the person is postioned relative to his/her starting point we need to add the two displacements. We draw a line ( the blue line ) from the starting point to the ending point of the journey, this is the resultant displacement and is labeled dr.

Addition Example
Q: Billy Bob Joe starts going to school from his house. First he walks north 300m to the bus stop, and then the bus takes him 1km west to school.

a) How far is Mr. Joe from his house when he is at school?
b) When Mr. Joe returns home what will his displacement be?
c) How will Mr. Joe's displacement be affected if he took a different route to school?

A: First lets draw our situation with vectors. Assuming north is up ( when in doubt draw the north arrow ) we draw Mr. Joe's northernly walk and then his westerly ride - a vector going up and a vector going left.

A triangle, the hypotenuse is resultant displacement The blue line is the resultant displacement - the final position which Billy Bob Joe has, the tail of the dr vector is at the house, and its head is at the school.

a) Our vector addition turned out to be a right angle triangle, and dr is the hypotenuse. Lets use the pythagorean theorem to find out just how far Mr. Joe is.
300² x 1000² = ||²
|| = 1044m
Mr. Joe is 1044 metres from his home.

Notice that we have used the absolute value notation || so that we can use the magnitude of without its direction.

b) When Billy Bob Joe returns home his displacement will be zero, he is zero metres away from his home.

c) Taking an alternate route would not change Mr. Joe's displacement, he would still end up in his school. Unlike distance, the displacement is not affected by how you got to a place, its only concerned with the location of that place.

Vector subtraction
You will very rarely use vector subtraction, although it is very simple: when subtracting a vector from another vector you are just adding an opposite vector.
1 - 2 = 1 + ( - 2 )
A vector multiplied by a negative number will have its direction reversed ( above we are multiplying by -1 so its magnitude stays the same ). Visually the vector which is being subtracted will flip its head and tail, then you rearrange the vectors head-to-tail again and perform simple addition.
Vectors and angles
While this section has been an introduction to vectors and slightly theoretical primer you should move on to the next chapter Vectors and Angles to see how useful vectors are when they are combined with angles.

The Basics - Simple Principles Of Motion

Distance is usually measured in metres ( SI units ) and is frequently represented by the variable d.
Time is usually measured in seconds and is represented by the variable t.
Speed is usually measured in metres per second ( m/s ) and is represented by v.
I use Vi and Vf to denote initial speed and final speed, respectively.
Remember that variables can be represented by any combination of letters, they are just names for quantities e.g. d = 55m, d = 0.3cm, potato = 43m, time-initial = 4s.

In all cases a movement is measured from a "reference point", if you measure the distance you walk to school from home you will measure from your house - which is the reference point in this case, it is the point from which you start measuring.

Standard International Units
When you are performing physics calculations you should always keep similar types of measurements in the same units. If you are performing a calculation with two masses which weigh 2kg and 2000g you should convert both of them into one unit type, either convert them into kilograms or convert them into grams. Do not mix units of measurement.

SI ( Standard International ) units are special conventions for measuring which simplify physics calculations considerably. The metre, second and kilogram are SI units. If you perform your calculation using the above standard units you will be able to refer to your answer's units by an alias, e.g. 88 kg/m/s² turns into 88 "Newtons".

Kinematics and motion lack space-saving shorthands for units you should nevertheless use SI units ( m, kg, s ) when practical. Whenever you do not use SI units make sure that your measurements are all in the units which you are using.

Basic Equations
Physics relies heavily on mathematics. When solving a problem involving physics you will need to convert words and events into mathematical concepts, this conversion results in an equation.

Speed Speed is the rate at which distance changes. How many distance units are passed during each unit of time?
speed = distance / time
v = d / t

Lets do an example:
A car travelled 50km in 120minutes while keeping a constant speed. What was its speed?

Before we start solving for the speed we need to convert our units into useable form. A car's speed will usually be represented in km/h, we need to convert the time of the trip into hours. We do this by dividing by 60, 120/60 = 2 hours.

Lets declare our variables
d = 50km
t = 2h
v = ? km/h

The speed is "how many kilometres are passed in one hour" and we come up with an equation to model that: v = d / t
We input our values into the equation:
v = 50 / 2
v = 25km/h

We always state the answer to the question in a sentence.
"The speed of the car during the trip was 25km/h".
Acceleration Acceleration is the rate of change in speed, how the speed differs from one moment to the next.
acceleration = speed / time
a = v / t

Also, the total acceleration undergone is widely thought of as the final speed - the initial speed, all divided by the time. This is a useful way to model acceleration because an object might have been already moving before it began accelerating.
a = (Vfinal - Vinitial) / t

Example
A car speeds up from rest up to 60km/h in 6.2 seconds, what is its average acceleration?

Convert all measurements into metres and seconds to prevent confusion and/or awkward result units.

t = 6.2s
Vinitial = 0m/s
Vfinal = 16.6m/s
a = ? m/s²

a = 16.6 / 6.2
= 2.677 m/s²

We need to round the answer to two Significant Digits ( what are significant digits? ) therefore our word answer will be "The car accelerates at the rate of 2.7 m/s²"

Physics Material

Lesson 1 Mechanics
Chapter Image 2. Velocity
Chapter Image 3. Acceleration
Chapter Image 4. Forces and Newton's Laws
Chapter Image 5. Motion in Two Dimensions
Chapter Image 6. Projectile and Periodic Motion
Special Chapter 7. Lesson 1 Review


Lights and Waves Lesson 2
Chapter Image 8. Waves
Chapter Image 9. Sound
Chapter Image 10. Light


Lesson 3 Electricity
Chapter Image 11. Electric Forces
Chapter Image 12. Electric Field
Chapter Image 13. The Current
Chapter Image 14. Basic Circuit
Special Chapter 15. Advanced Circuit

Basic Physics

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.


Check!

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.


Check!


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:


Check!


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



Check! 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:
Check!
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?