Kinematics and Particle Models

Particle Models

Particle models are simplified representations of objects in motion, treating them as point particles to analyze their movement without considering their size or shape.

• Constant motion: Represented by evenly spaced dots.

• Speeding up: Dots get farther apart as time progresses.

• Slowing down: Dots get closer together.

• Quadratic motion: Dots show increasing or decreasing spacing in a non-linear fashion.

• Measurement convention: The particle model does not consider the object's size; for example, measurement may start at the bottom of a rocket and end at the top.

Reference Frames and Arbitrary Choices

Directions and sign conventions in kinematics are chosen for convenience and can be arbitrary.

• Right: Often considered positive.

• Left: Often considered negative.

• Up: Often considered positive.

• Down: Often considered negative.

• Choices may be made differently to simplify calculations.

Motion and Kinematic Variables

Displacement and Position

Displacement is the change in position of an object, and is a vector quantity.

• Displacement ():

• Position (, ): Location at specific moments.

Velocity and Speed

Velocity is the rate of change of position and is a vector; speed is the magnitude of velocity and is a scalar.

• Velocity ():

• Speed:

Position and Velocity Graphs

Position and velocity graphs are related, but velocity graphs alone do not reveal the starting position.

• Position vs. Time: Shows how position changes over time.

• Velocity vs. Time: Shows how velocity changes over time.

• Key Point: You cannot determine the starting position just from velocity graphs.

Kinematic Variables and Units

Key Variables

• , : Position at specific moments

• : Change in position over an interval

• , : Instantaneous velocities

• : Change in velocity over an interval

• : Average velocity over an interval

• , : Moments in time

• : Change in time

• : Acceleration (assumed constant unless stated otherwise)

Units of Measurement

Acceleration and Its Interpretation

Definition and Formula

Acceleration is the rate of change of velocity with respect to time.

• Formula:

• Acceleration can be positive (speeding up) or negative (slowing down), depending on the direction of velocity and the sign convention.

Examples of Acceleration in Motion

• Forward, speeding up: Velocity increases above the x-axis ( acceleration).

• Forward, slowing down: Velocity decreases above the x-axis ( acceleration).

• Backward, speeding up: Velocity is negative and moving away from zero ( acceleration).

• Backward, slowing down: Velocity is negative and moving toward zero ( acceleration).

Kinematic Equations

Constant Velocity

When velocity is constant, displacement is given by:

Constant Acceleration

If acceleration is constant, the following kinematic equations apply:

Free Fall and Symmetrical Motion

Free Fall

Objects in free fall experience the same acceleration due to gravity, , when no other forces act (e.g., air resistance is neglected).

• Free fall: Only gravity acts on the object.

• Graph: Velocity decreases linearly with time, slope .

Symmetrical Motion

• Distance up equals distance down.

• Launch speed equals landing speed.

• Time going up equals time going down.

• Velocity is zero at peak motion.

Comparing Motions: Numeric and Algebraic Methods

Numeric Comparisons

Solve for the motion of each object independently, then compare results.

Algebraic Comparisons

Final moment is defined by a relationship between two objects (e.g., when they meet).

Example: Helicopter and Box

A helicopter rises steadily at while lifting a box hanging on a rope. At seconds after the rope breaks, how far below the helicopter is the box?

Distance below helicopter:

Area Under Curves: Displacement from Velocity Graphs

Calculating Displacement

The area under a velocity vs. time graph gives the displacement.

• Rectangle:

• Trapezoid:

Examples

• For a velocity graph with constant velocity, use the rectangle formula.

• For a graph with changing velocity, use the trapezoid formula to find total displacement.

Example Calculation: If , , , then .

Additional info: These notes cover foundational concepts in kinematics, including graphical analysis, equations of motion, and problem-solving strategies for one-dimensional motion.