Understanding Motion: Kinematics and Dynamics

Motion is the act or process of changing position or orientation in space and time.
It is a fundamental concept in physics, describing how objects move.
The study of motion is divided into kinematics and dynamics.
Kinematics describes the motion of objects without considering the forces that cause the motion.
Dynamics studies the relationship between the motion of objects and the forces acting on them.

Linear motion (or translational motion) occurs when an object moves along a straight line.
Examples include a car moving on a straight road or a ball dropped from a height.
Circular motion occurs when an object moves along a circular path.
Examples include a satellite orbiting the Earth or a car turning a corner.
Projectile motion is the motion of an object thrown or projected into the air, subject to gravity.
Examples include a ball thrown upwards or a bullet fired from a gun.
Rotational motion occurs when an object spins or rotates around an axis.
Examples include a spinning top or a rotating wheel.
Oscillatory motion is repetitive back-and-forth movement around an equilibrium position.
Examples include a swinging pendulum or a vibrating spring.

Deals with the description of motion, regardless of the forces causing it.
Includes concepts like displacement, velocity, and acceleration.
Equations of motion (for constant acceleration):
- v = u + at (final velocity = initial velocity + (acceleration * time))
- s = ut + (1/2)at² (displacement = (initial velocity * time) + (1/2 * acceleration * time²))
- v² = u² + 2as (final velocity² = initial velocity² + (2 * acceleration * displacement))
Where:
- v is the final velocity
- u is the initial velocity
- a is the acceleration
- t is the time
- s is the displacement

An object at rest stays at rest, and an object in motion stays in motion with the same speed and direction unless acted upon by a force.
Inertia is the tendency of an object to resist changes in its state of motion.
Mass is a measure of an object's inertia.

The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
Expressed mathematically as F = ma, where F is the net force, m is the mass, and a is the acceleration.
Force is a vector quantity, measured in Newtons (N).

For every action, there is an equal and opposite reaction.
When one object exerts a force on another object, the second object exerts an equal and opposite force on the first.
Action and reaction forces act on different objects.

Momentum is a measure of an object's mass in motion.
It is the product of an object's mass and velocity: p = mv, where p is momentum, m is mass, and v is velocity.
It is a vector quantity.
Conservation of Momentum: In a closed system, the total momentum remains constant if no external forces act on the system.
This principle is important in analyzing collisions.

Energy is the capacity to do work.
Measured in Joules (J).
Kinetic Energy (KE): The energy an object possesses due to its motion.
KE = (1/2)mv², where m is mass and v is velocity.
Potential Energy (PE): Stored energy due to an object's position or condition.
- Gravitational Potential Energy: PE = mgh, where m is mass, g is the acceleration due to gravity, and h is height.
Work: The transfer of energy when a force causes displacement.
W = Fd cosθ, where F is force, d is displacement, and θ is the angle between the force and displacement vectors.
Work-Energy Theorem: The work done on an object equals the change in its kinetic energy.

Angular Displacement (θ): The angle through which an object has moved on a circular path.
Measured in radians.
Angular Velocity (ω): The rate of change of angular displacement.
ω = Δθ/Δt, measured in radians per second (rad/s).
Angular Acceleration (α): The rate of change of angular velocity.
α = Δω/Δt, measured in radians per second squared (rad/s²).
Centripetal Acceleration: The acceleration directed towards the center of the circular path, necessary to keep an object moving in a circle.
ac = v²/r = rω², where v is the linear velocity and r is the radius of the circle.
Centripetal Force: The force that causes centripetal acceleration.
Fc = mv²/r = mrω².

Motion of an object thrown into the air and subject to gravity.
The path of a projectile is a parabola (in ideal conditions, neglecting air resistance).
Initial velocity can be broken into horizontal (vx) and vertical (vy) components.
Horizontal motion has constant velocity (neglecting air resistance).
Vertical motion is affected by gravity (constant acceleration downwards).
Range (R): The horizontal distance traveled by the projectile.
Maximum Height (H): The maximum vertical displacement of the projectile.
Time of Flight (T): The total time the projectile is in the air.

Torque (τ): A force that causes rotation.
τ = rFsinθ, where r is the distance from the axis of rotation to the point where the force is applied, F is the magnitude of the force, and θ is the angle between the force and the lever arm.
Moment of Inertia (I): A measure of an object's resistance to rotational motion.
Analogous to mass in linear motion.
Depends on the mass distribution of the object.
Rotational Kinetic Energy (KErot): The kinetic energy due to rotational motion.
KErot = (1/2)Iω².
Angular Momentum (L): A measure of an object's rotational momentum.
L = Iω.
Conservation of Angular Momentum: In a closed system, the total angular momentum remains constant if no external torques act on the system.