1. Kinematics: Describing Motion

1-1 Reference Point and Displacement

Understanding motion requires specifying a reference point, which serves as the origin for measuring position and displacement.

• Reference Point: The fixed point from which position is measured.

• Displacement: The change in position of an object; a vector quantity with both magnitude and direction.

• Equation:

• Positive/negative sign depends on chosen axis direction.

1-2 Average Velocity

Average velocity quantifies how fast and in what direction an object's position changes over time.

• Definition: Velocity = displacement / time.

• Uses displacement, not distance.

• Speed is distance / time (no direction).

• Equation:

• Positive velocity: motion in the positive direction.

• Negative velocity: motion in the opposite (negative) direction.

1-3 Instantaneous Velocity

Instantaneous velocity is the velocity at a specific instant, as shown by the slope of a position vs. time graph at a point.

• Equation:

1-4 Acceleration

Acceleration measures how quickly velocity changes over time.

• Definition: Acceleration = change in velocity / time.

• If velocity and acceleration are in the same direction, the object speeds up.

• If velocity and acceleration are in opposite directions, the object slows down.

• Equation:

• Units: m/s2

2. Motion with Constant Acceleration

2-1 Kinematic Equations

For motion with constant acceleration, the following equations describe position and velocity over time:

• Velocity after time :

• Position after time :

• Velocity-displacement relation (no time):

• Average velocity:

2-2 Solving Problems

To solve kinematics problems, follow these steps:

1. Draw a diagram.

2. Write down knowns/unknowns.

3. Choose the right equation.

4. Watch signs (+/-) based on chosen direction.

2-3 Freely Falling Objects

Objects near Earth's surface experience constant acceleration due to gravity.

• Acceleration due to gravity: downward.

• Upward motion: velocity decreases by each second.

• At the top: velocity = 0, acceleration still .

• Falling down: velocity increases downward at .

2-4 Graphical Analysis

Graphs are useful for visualizing motion and interpreting velocity and acceleration.

• Position vs. time: slope = velocity.

• Velocity vs. time: slope = acceleration; area = displacement.

3. Vectors and Two-Dimensional Motion

3-1 Vectors and Scalars

Physical quantities are classified as scalars or vectors.

• Scalar: Magnitude only (e.g., speed, mass, time, distance).

• Vector: Magnitude and direction (e.g., velocity, acceleration, displacement, force).

Vector Components:

Resultant Vector:

3-2 Addition of Vectors — Graphical Methods

Vectors can be added graphically using the tip-to-tail or parallelogram methods.

• Tip-to-tail method: Place vectors head-to-tail; resultant is from start to end.

• Parallelogram method: Draw vectors from the same point, complete parallelogram, diagonal = resultant.

3-3 Subtraction & Multiplying by a Scalar

• Subtraction: Reverse direction of the vector being subtracted.

• Multiplying by a scalar: Changes length (magnitude), may reverse direction if scalar is negative.

3-4 Adding Vectors by Components

• Break each vector into x- and y-components.

• Resultant vector: use Pythagorean theorem and trigonometry.

4. Projectile Motion

4-1 Basic Principles

Projectile motion involves two independent motions: horizontal and vertical.

• Horizontal motion: constant velocity.

• Vertical motion: constant acceleration due to gravity.

4-2 Initial Velocity Components

4-3 Equations of Motion

• Horizontal:

• Vertical:

• Velocity components over time:

(constant)

4-4 Solving Projectile Problems

1. Break velocity into components.

2. Use horizontal motion for time/distance.

3. Use vertical motion for height/time.

4. Combine results for full trajectory.

4-5 Parabolic Path

The path of a projectile (ignoring air resistance) is a parabola because vertical position depends on while horizontal position is linear in .

4-6 Relative Velocity

Relative velocity describes motion as seen from different reference frames.

• If reference frames are moving, use vector addition:

• Example: Boat crossing a river, plane in wind.

5. Graphical Analysis of Motion

5-1 Position vs. Time Graphs

Position vs. time graphs reveal information about velocity and acceleration.

• Slope = velocity.

• Curved slope = changing velocity (acceleration).

• Straight line = constant velocity.

• Flat (horizontal) line = velocity is zero.

• Tilted (nonzero slope) = constant nonzero velocity.

5-2 Example Graphs

6. Newton's Laws of Motion

6-1 Force

Force is a push or pull acting on an object, and is a vector quantity.

• Unit: Newton (N), where 1 N = 1 kg·m/s2

• Types: Contact forces (friction, tension, normal) and field forces (gravity, electric, magnetic).

6-2 Newton's First Law (Law of Inertia)

An object at rest stays at rest, and an object in motion continues in motion at constant velocity unless acted on by a net external force.

• No net force → no change in velocity.

• Inertia: The resistance of an object to changes in its state of motion; depends on mass.

6-3 Mass

• Mass: Amount of matter (scalar, kg).

• Weight: Gravitational force on mass ().

• Mass is a measure of inertia.

• Unlike weight, mass does not change with location.

6-4 Newton's Second Law

Net force causes acceleration; the direction of acceleration is the direction of net force.

• Equation:

• Units: N = kg·m/s2

6-5 Newton's Third Law

For every action force, there is an equal and opposite reaction force.

• Forces always occur in pairs acting on different objects.

• Example: You push on a wall, the wall pushes back with equal magnitude, opposite direction.

6-6 Weight and the Normal Force

• Weight:

• Normal force: Perpendicular contact force exerted by a surface.

• On a flat horizontal surface:

• On an incline:

6-7 Solving Problems with Newton's Laws: Free-Body Diagrams

1. Draw the object.

2. Represent all forces acting on it with arrows (length = magnitude, direction = direction of force).

3. Label each force clearly (gravity, normal, friction, tension, applied).

4. Write Newton's 2nd Law in component form (x and y axes).

Helps organize forces before solving acceleration, tension, friction, etc.

6-8 Problems Involving Friction, Inclines

• Friction: Opposes motion (or attempted motion).

• Static friction: (keeps object at rest).

• Kinetic friction: (acts on moving objects).

• Inclined plane: Break weight into components:

  • Parallel:

  • Perpendicular:

• Use these with Newton's 2nd Law to solve for acceleration, tension.