Kinematics: Scalars and Vectors in One Dimension Study Notes
Kinematics
Definition: Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces causing the motion.
Types of Motion:
Uniform motion: Constant velocity.
Non-uniform motion: Changing velocity (acceleration).
The Goal of a Physicist
To understand the fundamental workings of the universe.
Precisely Define Fundamental Quantities:
Identify and measure the most basic properties that describe the physical world.
Examples: time, distance, mass, velocity, and force.
These quantities serve as the building blocks for studying and explaining how the universe operates.
Discover Relationships Between Quantities:
Study how these fundamental quantities interact and influence one another.
Example: Newton's laws of motion describe how force, mass, and acceleration are interconnected.
These relationships allow physicists to develop models and equations that predict how objects and systems behave under various conditions.
Physical Quantities: Scalar vs Vector
Scalar:
A scalar quantity has magnitude only.
Examples: speed, mass, volume, time
Vector:
A vector has both magnitude and direction.
Examples: velocity, weight, friction
Quantities in Physics
Scalars:
Quantities that have only magnitude (a numerical value).
Sufficient to describe the size or amount of something but do not provide any information about direction.
Examples:
Distance (e.g., 10 meters)
Speed (e.g., 50 kilometers per hour)
Time (e.g., 5 seconds)
Mass (e.g., 20 kilograms)
Vectors:
Have both magnitude and direction, making them essential for describing quantities where direction is significant.
Examples:
Displacement (e.g., 10 meters to the north)
Velocity (e.g., 50 kilometers per hour to the east)
Acceleration (e.g., downward)
Force (e.g., 30 newtons acting at an angle of ).
Vectors are represented with arrows in diagrams, where the length of the arrow indicates the magnitude, and the direction of the arrow shows the quantity's direction.
Scalar vs Vector Examples
Distance: "The car traveled 100 kilometers."
Displacement: "The car moved 50 meters to the north."
Speed: "The plane is flying at 600 kilometers per hour."
Time: "The experiment lasted for 5 minutes."
Velocity: "The bike is traveling at 20 meters per second to the east."
Acceleration: "The object is accelerating at downward (free fall)."
Mass: "The bag weighs 10 kilograms."
Force: "A 30-newton force is applied at an angle of ."
Temperature: "The temperature today is ."
Momentum: "The ball has a momentum of 15 kg·m/s to the right."
Length of a car: 4.5 m (magnitude, physical quantity)
Time: 12.76 s (magnitude, physical quantity)
Mass of gold bar: 1 kg (magnitude, physical quantity)
Temperature: (magnitude, physical quantity)
Position of California from North Carolina: 3600 km in west (magnitude and direction, physical quantity)
Displacement from USA to China: 11600 km in east (magnitude and direction, physical quantity)
Vectors in Detail
Vectors are quantities with magnitude and direction
Because vectors include direction, they can be more complicated to work with than scalars
Vector Notation:
When vector quantity symbols are typed, they are sometimes written in bold to distinguish them from scalar quantities.
For example, the symbol v may be used for velocity (a vector) while the symbol v is used for speed (a scalar).
To indicate that a quantity is a vector, we use vector notation: we draw an arrow over the symbol for that quantity.
When all vector quantities in a given scenario are in the same dimension, their directions can be expressed with a "+" or "-" sign.
In order for the signs to have physical meaning, though, there must be a defined coordinate system stating which direction is assigned as positive and which as negative.
A commonly used coordinate system:
Left is designated as the negative direction and right is designated as the positive direction.
Drawing Vectors
A vector is like an arrow that shows both:
How big something is (magnitude).
Which direction it’s going.
To represent a vector, you draw an arrow:
The length of the arrow shows the size (magnitude) of the vector.
The direction the arrow points shows where it’s going.
Example: If you have a velocity of:
5 m/s to the right → draw a short arrow pointing right.
10 m/s to the left → draw a longer arrow pointing left.
Key Parts of a Vector:
The tip (or head): The pointy end of the arrow.
The tail: The opposite end of the arrow.
Example: If the arrow points up, the pointy end (tip) shows the direction, and the base (tail) is where it starts.
What About Scale?
If you’re comparing two vectors:
Make longer arrows for bigger magnitudes.
Make shorter arrows for smaller magnitudes.
Example: A 10 m/s vector should be twice as long as a 5 m/s vector.
Can We Move Vectors?
Yes!
You can shift a vector around as long as:
The length and direction stay the same.
Example: Four arrows of the same length and angle (all pointing up and right) represent the exact same vector, no matter where they’re drawn.
Understanding the Example
Coordinate System:
Imagine a line that goes left and right.
We call the right side positive (marked as +x), and the left side negative (marked as -x).
Velocity Vector:
A velocity vector is like an arrow that shows:
How fast something is moving (magnitude).
Which direction it's moving (to the left or right in this case).
Adding Vectors Visually
To add vectors visually, place the tail of the second vector at the tip of the first vector.
Draw a new vector from the tail of the first vector to the tip of the last vector; This new vector is the sum, or resultant, of the vectors being added.
The order in which vectors are added does not matter; the resultant vector will be the same.
Conclusion
Understanding kinematics and the concept of physical quantities (scalars and vectors) is essential in physics, as they form the foundation for analyzing motion and forces acting upon objects. By studying these concepts, physicists can create precise models and predictions about the behavior of different physical systems.