Chapter 3: Kinematics in 2 or 3 Dimensions, Vectors

3.1 Vectors and Scalars

Physical quantities in physics are classified as either vectors or scalars. Understanding the distinction is fundamental for analyzing motion in multiple dimensions.

• Vector: A quantity with both magnitude and direction. Examples include displacement, velocity, force, and momentum.

• Scalar: A quantity with only magnitude. Examples include mass, time, temperature, and speed.

• Notation: Vectors are denoted as \( \vec{A} \) in print and A when handwritten.

Example: The distance a car travels (scalar) versus its displacement from the starting point (vector).

3.2 Addition of Vectors—Graphical Methods

Vectors can be added graphically, which is especially useful for visualizing motion in one or two dimensions.

• In one dimension, simple addition and subtraction suffice, but attention must be paid to the signs (direction).

• In two dimensions, vectors at right angles can be added using the Pythagorean Theorem:

• Order of addition does not affect the resultant:

• For non-perpendicular vectors, use the tail-to-tip method or parallelogram method for graphical addition.

Example: Adding two displacement vectors at right angles to find the total displacement.

3.3 Subtraction of Vectors, and Multiplication of a Vector by a Scalar

Vector subtraction and scalar multiplication are essential operations in vector analysis.

• Subtraction: To subtract from , add the negative of :

• Multiplication by a Scalar: has the same direction as if , and the opposite direction if .

Example: Scaling a velocity vector by 1.5 or -2.0 changes its magnitude and possibly its direction.

3.4 Adding Vectors by Components

Any vector can be decomposed into perpendicular components, typically along the x and y axes. This method simplifies vector addition and is crucial for problem-solving in physics.

• Express as .

• Use trigonometric functions to find components:

• Components are added arithmetically since they are one-dimensional.

Steps for Adding Vectors by Components:

1. Draw a diagram and add vectors graphically.

2. Choose x and y axes.

3. Resolve each vector into x and y components.

4. Calculate each component using sines and cosines.

5. Add the components in each direction.

6. Find the length and direction using:

Example: A mail carrier drives 22.0 km north, then 47.0 km at 60° south of east. Find the total displacement using vector components.

3.5 Unit Vectors

Unit vectors are vectors of magnitude 1, used to specify direction along coordinate axes.

• Common unit vectors: \( \hat{i} \) (x-axis), \( \hat{j} \) (y-axis), \( \hat{k} \) (z-axis).

• Any vector can be written as:

3.6 Vector Kinematics

Kinematics in two or three dimensions involves analyzing displacement, velocity, and acceleration as vectors.

• Displacement:

• Average velocity:

• Instantaneous velocity:

• Instantaneous acceleration:

• Using unit vectors:

• Generalized equations for constant acceleration in two dimensions:

3.7 Projectile Motion

Projectile motion describes the motion of an object moving in two dimensions under the influence of gravity. The path is a parabola.

• Horizontal and vertical motions are analyzed separately.

• Horizontal speed is constant; vertical motion has constant acceleration downward.

• If launched at angle with initial speed :

• Kinematic equations for projectile motion:

In x-direction:

In y-direction:

Example: Calculating the trajectory of a baseball hit at an angle, determining if it clears a fence.

3.8 Solving Problems Involving Projectile Motion

Solving projectile motion problems involves systematic steps:

1. Read the problem and identify the object(s) to analyze.

2. Draw a diagram.

3. Choose an origin and coordinate system.

4. Decide on the time interval (object in motion under constant ).

5. Examine x and y motions separately.

6. List known and unknown quantities. Note is constant, at the highest point.

7. Plan the solution using appropriate equations.

Example Problem: A baseball is hit at m/s, , from m, toward a wall m away and m high. Find if the ball clears the wall.

• Calculate and .

• Find time to reach the wall: .

• Find at : .

• If , the ball clears the wall.

Range of Projectile: Maximum range when ():

Projectile Path Equation: This is the equation of a parabola.

3.9 Relative Velocity

Relative velocity describes how the velocity of an object appears from different reference frames. In two dimensions, velocities are added or subtracted as vectors.

• Each velocity is labeled by the object and the reference frame.

• Relationship:

Example: A boat must head upstream at an angle to compensate for river current to travel directly across.

Summary Table: Scalar vs. Vector Quantities

Summary of Chapter 3

• A quantity with magnitude and direction is a vector.

• A quantity with magnitude but no direction is a scalar.

• Vector addition can be done graphically or by components.

• The sum is called the resultant vector.

• Projectile motion is the motion of an object near Earth's surface under gravity.

Additional info: These notes expand on the original slides by providing full definitions, step-by-step procedures, and explicit equations in LaTeX format for clarity and completeness.