Chapter 1: Vectors (Incomplete)

INTRODUCTION

-Vectors show up everywhere in physics.

-Some physical quantities that are represented as vectors are: displacement, velocity, acceleration, force, momentum, and electric and magnetic fields.

-For now, we will study two-dimensional vectors (flat plane)

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DEFINITION

Vector- a quantity that involves both magnitude and direction and obeys the commutative law of addition.

Scalar- a quantity that does not involve in direction.

EX: 55 miles per hr is scalar, while 55 miles per hr, to the north is a vector

Other examples of scalars: mass, power, temperature, and electric charge

Vectors can be denoted in several ways:

A, A, or an A with an arrow on top

the two on the left are for textbooks while the one on the right is used in handwritten documents

Displacement (net distance traveled plus direction) is the prototypical example of a vector:

A= 4 miles to the north

displacement magnitude direction

B= 3 miles to the east

displacement magnitude direction

When we say that vectors obey the commutative law for addition, we mean that if we have two vectors of the same type, then A+B must equal B+A. The vector sum A+B means the vector A followed by B, while the vector sum B+A means the vector B followed by A.

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VECTOR ADDITION (GEOMETRIC)

Vectors are drawn as arrows. The tail is where the vector originates and the tip is where the arrow is. The sum of two vectors is expressed in a "tip-to-tail" method. (as seen in the above picture)

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SCALAR MULTIPLICATION

A vector can be multiplied by a scalar (a #), and the result is a vector. If the original vector is A and the scalar is k, then the scalar multiple kA is as follows:

magnitude of kA= |k| x (magnitude of A)

direction of kA= the same as A if k is positive OR the opposite of A if k is negative

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VECTOR SUBTRACTION (GEOMETRIC)

To subtract one vector form another, for example, to get A-B, simply form the vector -B, which is the scalar multiple (-1)B, and add it to A:

A - B = A + (-B)

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Chapter 2: Kinematics (Incomplete)

INTRODUCTION

-Kinematics is the study of an object;s motion in terms of its displacement, velocity, and acceleration.

Questions pertaining to this study are:

-How far does this object travel?

-How fast and in what direction does it move?
-At what rate does its speed change?

In the next chapter, I will study dynamics, which will explain in detail why objects move the way they do.

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POSITION

Any object exists in some part of the universe. We call that location the object's location, but without a reference point, the position said is meaningless. That being said, we can arbitrarily chose one location and call it the origin (like the origin in a Cartesian coordinate system). Usually, it is logical to let the object start at the origin, but it is not required because the laws of physics work everywhere. As we go more in depth, we will find numerous cases where we can greatly simplify a problem by manipulating the coordinate systems and its origin.

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DISPLACEMENT

Displacement is an object's change in position. It's the vector that points form the object's initial position to its final position, regardless of the path actually taken. Since displacement means change in position, it is generically denoted as Δs, where Δ denotes change in and s means spatial location. (The letter p is not used because it is used as the quantity of momentum.)

-If the displacement is horizontal, then it can be called Δx

-If the displacement is vertical, then it can be called Δy

The magnitude of this vector is the net distance traveled; sometimes the word displacement refers just to scalar quantity. Since a distance is being measured, the SI unit for displacement is the meter [Δs] = m.

EX:

Q: A rock is thrown straight upward from the edge of a 30 m cliff rising 10 m then falling all the way down to the base of the cliff. Find the rock's displacement.

A: Since displacement only refers to the object's initial position and final position, not the details of its journey. Therefore, its displacement is 30 m, downward (from the edge to the bottom).

EX: look in pg 11 for the picture

Q: An infant crawls 5 m east, then 3 m north, then 1 m east. Find the magnitude of the infant's displacement.

A: Although the infant crawled a total distance of 5 + 3 + 1 = 9 m, this is not displacement, which is merely thenet distance traveled.

Using the Pythagorean theorem, we can calculate that the magnitude of the displacement is

Δs = √[(Δx)^2 + (Δy)^2] = √[(6 m)^2 + (3 m)^2] = √(45 m^2) = 6.7 m

EX:

Q: In a track-and-field event, an athlete runs exactly once around an oval track, a total distance of 500 m. Find the runner's displacement of the race.

A: If the runner returns to the same position from which she left, then her displacement zero.

***A note about Notation***

Δs is a more general term that works in space. The term x or "Δx = xf - xi" (x final minus x initial) has a specific meaning that is defined in the x direction. However to be consistent with AP notation, and to avoid confusion between spatial location and speed, from this point on we will use x in our development of the concepts of speed, velocity, and acceleration. The concepts work the same in y direction.

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SPEED AND VELOCITY

When in a moving car, th speedometer tells us how fast we are going. For example, a car may be going at a speed of 50 mph in an expressway. Average speed is the ratio of the total distance traveled to the time required to cover that distance:

average speed = total distance/time

However, the car's speedometer doesn't care in what direction the car is moving, whether I am driving 55 mph north, south, east, or west; it is still 55 mph. Speed is scalar.

However, in physics, it is very important to note what direction the object in question is moving. We just learned about displacement, which takes both distance (net distance i to f) and direction into account. The single concept that takes both speed and direction is called velocity, and the definition of average velocity is:

average velocity = displacement/time

v = Δx/Δt

(The bar over the v means average.) Because Δx is a vector, and because Δt is a positive scalar (b/c there is no such thing as negative time), the direction of v is the same as the direction of Δx. The magnitude of the velocity vector is called the object's speed, and is expressed in units of meters per seconds (m/s).

Note the distinction between speed and velocity. In everyday language, they're often used interchangeably. However they are not in physics. Velocity is speed plus direction.

***A note about velocity and speed***

The magnitude of velocity is speed. However (this is a bit confusing), the magnitude of the average velocity is notcalled the average speed. Average speed is defined as the total distance traveled divided by the elapsed time. One the other hand, the magnitude of the average velocity is the net distance traveled divided by the elapsed time.

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EX:

Q: An infant crawls 5 m east, then 3 m north, then 1 m east. If the infant completes his journey in 20 seconds,find the magnitude of his average velocity.

A: Since his displacement is 6.7 m, the magnitude of his average velocity is

v = Δx/Δt = (6.7 m)/(20 s) = 0.34 m/s

Chapter 3: Newton's Laws

Chapter 5: Linear Momentum

Chapter 4: Work, Energy, and Power

Chapter 6: Newton's Law of Gravitation

Chapter 7: Oscillations

Chapter 8: Fluid Mechanics

Chapter 9: Thermal Physics

Chapter 10: Electric Forces and Fields

Chapter 11: Electric Potential and Capacitance

Chapter 12: Direct Current Circuits

Chapter 13: Magnetic Forces and Fields

Chapter 14: Electromagnetic Induction

Chapter 15: Waves

Chapter 16: Optics

Chapter 17: Atomic and Nuclear Physics