 # Kinematics Chapters 2 & 3.

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Kinematics Chapters 2 & 3

Definitions Physics – the science that deals with the relationship between matter and energy. Kinematics – the study of how objects move

There are 2 types of measurements in everyday life:
Scalar – only have size (magnitude). Ex. Mass, time, energy, distance, speed Vectors – have a size and direction. Ex. Weight, velocity, displacement, acceleration.

Motion Frame of reference – a physical part of the world, defined by an observer, which is used to discuss motion or compare motions. Distance – how far an object has traveled; a scalar; d Position – describes where an object is relative to the frame of reference; a vector;

Displacement – the change in an object’s position as it moves from point A to point B; a vector;
Speed – the distance traveled over a certain period of time; a scalar; v Velocity – how fast and in what direction; rate of change of position; a vector; Average velocity or

Clock reading – instantaneous; exact time you see when you look at a clock; a scalar; t
Time interval – elapsed time between 2 instants of time; a scalar; Δt

Summary Scalar (size only) Vector (size & direction) Mass Time Energy
Distance Speed Clock reading Vector (size & direction) Weight Displacement Acceleration Velocity Position

Vectors Vectors: A quantity with a direction as well as a magnitude
Graphically represented by an arrowed line segment with a length equal to its magnitude (drawn to scale) and the direction of the arrow indicating the direction of the quantity. Drawn tail to head. Often represented in bold, italic, typed letters ( ) without arrows but when using them in addition, etc. you must show they are vectors

Vector Addition Involves adding vectors together to obtain a resultant vector ( ) The resultant vector is a single vector that could represent the same action as adding vectors and

Vectors are always joined in the following way: The tail of one vector is placed at the head of another vector. Resultants are shown using dotted lines and are drawn from the tail of the first vector to the head of the last vector. The order of addition doesn’t matter, but you cannot change the size or direction of the vectors.

There are 2 methods of vector addition:
Graphical (drawn to scale, use ruler and protractor) Analytical (trig or Pythagorean Theorem for 2D)

You must follow the rules for adding vectors (given above) in each case below: 1. Adding vectors in the same dimension Draw to scale Order of addition doesn’t matter you can move the vectors freely through space, but you can NEVER change the direction or magnitude of the vector.

2. Adding vectors in 2 dimensions:
Measure the magnitude of the resultant using a ruler and your scale Measure the direction of the resultant using a protractor... In math books the angle is measured counterclockwise from the horizontal. We measure from the starting point to the resultant usually in reference to the nearest x-axis. It is reported as [E60oN] which is the same as [N30oE]. 3. Several vectors If you have 3 or more vectors, find the resultant by placing vectors tail-to-head. Use ruler, scale, and protractor to solve.

Practice Problems 1. Joe walks 10 m [E], then 40 m [E]. Graphically find the displacement. Scale 1 cm = 10 m, let [E] be positive. 2. Shari walks 10 m [E], then 40 m [W]. Graphically find the displacement. Scale 1 cm = 10 m, let [E] be positive.

3. Jim walks 10 m [E], then 40 m [N] and 5 m [W]
3. Jim walks 10 m [E], then 40 m [N] and 5 m [W]. Graphically find the displacement. Do Practice Problems #s 1-3 Page 112 Red Book

Perpendicular Vectors
Perpendicular vectors quantities are independent of each other. Ex. If a boat is traveling 8.0 m/s [E] across a river that flows north at 5.0 m/s. This means that in one second, the boat travels 8.0 m east as well as 5.0 m north. The velocity north (river’s velocity) does not change the velocity east (boat’s velocity) or vice versa!

Perpendicular vectors can be added to get a resultant velocity.
The resultant of the previous example would be 9.4 m/s [N58oE]. (Start at North axis and move 58o to the East.) This means the same as saying in one second, the boat travels 8.0 m east and 5.0 m north at the same time.

The graphical method can be time consuming and require a lot of space. We can use numbers to represent vectors instead of arrows. Numbers can represent size and direction. When vectors are in the same plane, the vectors can be added or subtracted. East and North are positive directions. South and West are in the negative direction.

You can find the resultant of 2 perpendicular vectors using the Pythagorean Theorem
(c2 = a2 + b2) and trigonometry which is the branch of math that deals with the relationship between the sides and angles in triangles.

Trig Functions The trig functions we will use are: Sin Θ = opp/hyp

When you have 2 non-perpendicular vectors or 3 vectors you can still find the resultant using vector resolution.

Practice Problems Do Practice Problems #s 1-3 on Page 112 Red Book
Do Example Problem Page 115 Red Book Do Practice Problems #s 4-9 Pages 112 – 116 Red Book

Vector Resolution So far, we have added 2 or more vectors to obtain a resultant which is a single vector with the same effect as the original or component vectors. Now, we are going to go backwards and start with a vector and think of it as the resultant of 2 vectors. Vector resolution is the process of finding the size of a component in a given direction.

Vector resolution can be done using the graphical method of vector addition or the analytical method. Choose axes according to information given in problem. The sign of the components can be found from the vector diagram. East and North are positive directions. West and South are negative directions.

Note: [N20oE] means start at north axis and move 20o to the east
Note: [N20oE] means start at north axis and move 20o to the east. Same measurement as 70o when measuring from x axis. Do Example Problem on Page 118 (Red Book) Do Practice Problems on Page 118 #s 11-14 Vector Worksheet

Describing Motion (Ch 2)
Motion is common to everything and it can be described in three ways: 1. Words 2. Equations 3. Graphs When we study motion we will only describe it, not seek to explain its causes. We will look at objects that move in only straight lines and we will study objects as if they were point objects rather than 3-D ones.

Straight Line Motion Free falling objects exhibit straight line motion
Many objects move in complicated motions that can be considered combinations of 2 or more straight line motions.

Velocity Instantaneous velocity – is the velocity of an object at one moment in time. Constant velocity (or uniform motion) – is the velocity of the object is not changing, it remains the same, constant, or uniform. (A change in direction is considered changing the velocity.) Most objects do not move with a constant velocity ( ) as the speed and direction must always remain the same. Objects either speed up or slow down and change direction depending on obstacles or barriers.

Constant Velocity – This formula can be rearranged to find displacement from velocity and time.

Position-Time Graph A position-time graph shows how the position of an object depends upon time. Thus, time is the independent variable (x axis) and position is the dependent variable (y axis).

Since the graph is a straight line, we know that there is a linear relationship between time and position or that the object is traveling at a constant velocity. From the graph, you can get the slope of the line (slope = rise/run or Δy/Δx). The slope of the straight position-time graph is the constant velocity of the object because Δy is and Δx is !

If the position time graph is not a straight line, then draw a line of best fit to find the slope. The slope between the 2 points will be the average velocity between those points.

Average velocity: The average velocity will be positive for an object speeding up. The average velocity will be negative for an object slowing down.

A straight horizontal line on a PT graph shows an object at rest
A straight horizontal line on a PT graph shows an object at rest. (No motion)

Interpreting Position Time Graphs
Describe each line for speed and position.

Describe a scenario to explain this position time graph.

Do Model Problems 1 and 2 on Pages 42-45.
Do Problems 1-3 Page 45-46 Chapter 3 Worksheet (Red Book) Concept Review and Practice Problem Worksheet (Red Book)

See Concept Organizer on Page 58
The 3 Position Time graphs show the difference between constant, average, and instantaneous velocity.

Dot Diagrams Read Pages 31-32 Dots can be used to represent objects.
This simplifies the point you measure from. It is as if you are concentrating the entire object into that one dot. To use dot diagrams, something has to be consistent, Figures 2.3 and 2.4 use equal time intervals. Do Section 2.1 Review on Page 34 #s 1-3 Do Conceptual Problem #2 on Page 40.

Instantaneous Velocity
Velocity changes if either the speed or direction of the object changes. Recall average velocity or uniform velocity produce a straight line on a position time graph. Do Model Problem on Page 55 Do Problems 4 and 5 on Pages 57-58

Velocity Time Graphs Velocity Time Graphs describe motion with either a constant or a changing velocity. The area under the curve of a velocity time graph is the displacement of the object from its original position at some time, t.

Describe the motion in the following Velocity Time Graph and calculate the displacement.

To calculate the instantaneous velocity from a VT Graph that is not uniform motion (curve) find the slope at the tangent.

Relativity of Motion Motion is relative which means that an object can be moving with respect to one body and at the same time, be at rest or moving at a different velocity with respect to a second body. Ex. You are driving past a house in a car. Ex. Your are driving past a cyclist and a house in a car. When measuring either position or velocity, you must first define your frame of reference.

Do Sample Problem Page 56 (Red Book)
Do Practice Problem #20 Page 56 (Red Book)

Do Quick Lab Page 59

Do Practice Problems #s 13-19 Page 60 (Red Book)
Do #27 Page 130 (Red Book)