Momentum, Energy, Collisions and Projectile Motion

• These notes cover momentum, energy, collisions and projectile motion.

Momentum

• Defined as the product of mass and velocity; also known as inertia in motion.
• Represented as p = mv, with units of kg.m/s.
• Mass can be derived from momentum and velocity: m = p/v (kg).
• Velocity can be derived from momentum and mass: v = p/m (m/s).

Impulse

• Defined as the product of force and time; it causes a change in momentum, represented as I = Ft = Δρ.
• Impulse is measured in Newton-seconds (N.s).
• Force equals mass multiplied by acceleration: F = ma (N).
• Acceleration is the rate of change of velocity: a = F/m (m/s²).

Isolated System

• A system with no external force acting upon it.
• An external force changes an object's momentum.

Law of Conservation of Momentum

• In the absence of an unbalanced external force, the linear momentum of a system remains constant.
• Momentum conservation applies to billiard balls colliding on a table.
• Momentum conservation also underpins rocket propulsion, where the momentum of the rocket equals the momentum of the expelled gases.

Collisions

• Interactions where momentum is exchanged or transferred and classified as elastic or inelastic.

Elastic Collisions

• Colliding bodies separate after impact.
• Both momentum and kinetic energy are conserved.
• Billiard ball collisions exemplify elastic collisions.

Inelastic Collisions

• Colliding bodies stick together after impact.
• Only momentum is conserved.
• Kinetic energy loss occurs through vibration, heat, or material distortion.

Momentum Conservation Formula

• m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂' where m1,2 are masses, v1,2 are initial velocities, and v1',2' are final velocities.
• In a system with no external forces, total momentum before a collision equals total momentum after.

Potential Energy

• The energy an object has while at rest.
• Defined operationally as PE = mgh, where m is mass, g is gravity, and h is height, measured in joules (J).
• Related to the force altering the mass's position.

Gravitational Potential Energy (GPE)

• The work performed against gravity's force, given by GPE = mgh.

Elastic Potential Energy (EPE)

• Reflects how much a body is stretched or compressed, calculated by EPE = 1/2 kx^2, where k is the spring constant and x is displacement.

Kinetic Energy

• Energy possessed by an object in motion, with KE = ½ mv^2.

Mechanical Energy (ME)

• The sum of energy of a body due to its motion and position, calculated by ME = PE + KE.
• ME is linked to an object's position and motion, resulting in work.

Work

• The effective force applied parallel to the resulting displacement, calculated by W = Fd, in joules (J) or (N · m).

Law of Conservation of Energy

• Energy cannot be created or destroyed, only transformed.
• The total initial energy equals the total final energy in a closed system: Ei = Ef.

Displacement

• Shortest distance between initial and final points.

Velocity

• Speed in a specific direction.

Projectile Motion

• When any object is thrown in space under the influence of gravity.
• Can be horizontal or at an angle.

Motion

• When an object changes position with respect to a reference point.

Distance

• The length along a path.

Speed

• The distance traveled per unit of time, denoted as 'v'.

Acceleration

• The rate of change in velocity, denoted as 'a'.

Horizontal Projectile

• Exhibits horizontal motion in the direction of the initial force and also has vertical motion due to gravity.

Uniform Horizontal Motion

• Constant horizontal velocity (vx = v1).
• Zero horizontal acceleration (ax = 0 m/s^2).

Accelerating Vertical Motion

• An acceleration due to gravity (ay = 9.8 m/s^2).

Components of Projectile Motion

• Includes height (vertical distance) and range (horizontal distance).

Projectile at an Angle

• Object motion when thrown from the ground at an angle to the horizontal.
• Follows a symmetrical, curved trajectory without air resistance.

Horizontal Component

• The horizontal velocity (vx) remains constant, given force is only applied initially.

Vertical Component

• Vertical velocity decreases moving upwards and reaches zero at max height due to gravity, -9.8 m/s².
• Vertical speed increases with gravity at 9.8 m/s².

Velocity (Components)

• Projectile motion velocity includes horizontal and vertical components.

Notes covering momentum, energy, collisions, and projectile motion. Momentum is the product of mass and velocity. Impulse is the product of force and time and causes a change in momentum. The notes also cover isolated systems and the law of conservation of momentum.