Vectors in Physics

Scalar Multiplication of Vectors

Multiplying a vector by a scalar changes its magnitude and possibly its direction, depending on the sign of the scalar. This operation is fundamental in physics for scaling quantities such as velocity, force, and displacement.

• Full Vectors:

• Units: Scalar may have units (e.g., kg, s, etc.), which can change the physical meaning of compared to .

• Magnitude:

• Direction: if ; if

• Components: If , then

Examples:

• Multiplication by : Stretches the vector (makes it longer)

• Multiplication by : Shrinks the vector (makes it shorter)

• Multiplication by : Reverses the direction

• Multiplication by $0$: Results in a zero-length vector (no direction)

Vector Addition

Vectors can be added to find a resultant vector, which represents the sum of the individual vectors. This is essential for analyzing forces, velocities, and other vector quantities in physics.

• Full Vectors:

• Graphical Method:

  1. Draw vector

  2. Draw vector starting from the tip of (tip-to-tail method)

  3. Draw the resultant vector from the start of to the end of

• Adding Multiple Vectors:

• Commutativity: Vector addition is commutative:

Frames of Reference and Relative Velocity

Frame of Reference

A frame of reference is the perspective from which an observer measures positions, velocities, and other physical quantities. It is often denoted with its own axes and origin.

Relativity and Relative Velocity (Galilean Relativity)

Relative velocity describes how the velocity of an object appears from different frames of reference. In classical (Galilean) relativity, velocities add linearly.

• Relative Velocity Equation:

• More Compact Notation:

• This holds in 1-D, 2-D, or 3-D.

• To reverse the order of frames, flip the direction:

Example: If a train moves at east and a person walks at east inside the train, their velocity relative to the ground is east.

Kinematics: Position, Velocity, and Acceleration

Instantaneous Position, Velocity, and Acceleration

Kinematics describes the motion of objects using position, velocity, and acceleration as functions of time.

• Position: or

• Velocity: or

• Acceleration: or

• Instantaneous Speed:

Equations of Motion (Constant Acceleration)

• General:

• Horizontal Components Only:

• Vertical Components Only:

• Freefall (choosing upward, downward):

  • or (if no gravity)

Projectile Motion

Projectile motion combines horizontal and vertical motion under gravity, resulting in a parabolic trajectory.

• Path Equation:

• Range (for level ground):

• Range is maximized when

• For any achievable range (between $0R_{max}\theta_0

• Adding air resistance/drag reduces max height, makes the path asymmetrical, and reduces range

Forces in Mechanics

Definition of Force

A force is any physical "push" or "pull" exerted on an object. It is a vector quantity, meaning it has both magnitude and direction.

Common Forces in Mechanics

Mass vs. Weight

• Mass: Amount of matter in an object (measured in kilograms [kg] or grams [g])

• Mass causes inertia (resistance to acceleration)

• Mass causes an object to respond to gravitational force

• Weight: Gravitational force acting on mass ()