CONSERVATION OF MOMENTUM | 5.1

Momentum

A measure of the quantity of motion of an object; depends on both mass and velocity and has the same direction as velocity.

Formula for Momentum

"p = m*v; where p is momentum

Units of Momentum

kg·m/s or N·s.

Nature of Momentum

A vector quantity; direction is the same as the object’s velocity.

Newton’s Second Law (Momentum Form)

The net force on an object equals the rate of change of its momentum: F_net = dp/dt.

Meaning of Momentum Form of Newton’s Second Law

"Momentum changes when a net external force acts; if net force is zero

Impulse

The product of force and the time interval during which the force acts; represents a change in momentum.

Formula for Impulse (Constant Force)

J = F_avg * Δt.

Formula for Impulse (Variable Force)

J = ∫ F dt; equal to the area under the Force–Time graph.

Units of Impulse

N·s (same as momentum).

Impulse–Momentum Theorem

The impulse applied to an object equals its change in momentum: J = Δp = pf - pi = F_net * Δt.

Physical Meaning of Impulse–Momentum Theorem

A large change in momentum can result from a large force acting briefly or a smaller force acting longer.

Airbags and Seatbelts Concept

"Increase collision time (Δt)

Work–Energy vs. Impulse–Momentum

"Work changes kinetic energy (distance-based)

Conservation of Momentum

"In an isolated system (no external forces)

Isolated System

A system with no net external forces acting on it; internal forces cancel out.

Why Momentum is Conserved

A direct consequence of Newton’s Third Law (action–reaction forces are equal and opposite).

Total Momentum of a System

The vector sum of all individual momenta: P = Σ(mi * vi).

Momentum Components

"Momentum is conserved separately in the x

Collision

"An interaction between two or more objects where large forces act for a short time

Elastic Collision

A collision in which both momentum and kinetic energy are conserved.

Inelastic Collision

"A collision in which momentum is conserved but kinetic energy is not (some energy is lost as heat

Perfectly Inelastic Collision

A type of inelastic collision in which the colliding objects stick together and move as one mass after impact.

Formula for Perfectly Inelastic Collision

(m1 + m2)vf = m1v1i + m2v2i; vf = (m1v1i + m2v2i)/(m1 + m2).

Example of Perfectly Inelastic Collision

A bullet embedding in a wooden block (ballistic pendulum).

Ballistic Pendulum Concept

Uses conservation of momentum during collision and conservation of energy during the swing to find a projectile’s speed.

Elastic Collision Equations (1D)

Momentum: m1v1i + m2v2i = m1v1f + m2v2f; Kinetic energy: (1/2)m1v1i² + (1/2)m2v2i² = (1/2)m1v1f² + (1/2)m2v2f².

Relative Velocity in Elastic Collision

The relative speed before and after collision has the same magnitude but opposite direction: v1i - v2i = -(v1f - v2f).

Elastic Collision (Equal Masses)

"If m1 = m2

Elastic Collision (Heavy–Light Case)

"If m1 >> m2

Elastic Collision (Light–Heavy Case)

"If m1 << m2

Two-Dimensional Collisions

Momentum is conserved independently in the x and y directions.

Perfectly Inelastic Collision (2D)

The objects stick together and move with a common velocity in a direction found using vector addition.

Center of Mass (CM)

The point representing the average position of all the mass in a system; acts as if all mass were concentrated there.

Formula for Center of Mass (Two Objects)

x_cm = (m1x1 + m2x2)/(m1 + m2).

Formula for Center of Mass (Many Objects)

"xcm = Σ(mi xi)/Σ(mi)

Physical Meaning of Center of Mass

The balance point or 'average location' of mass distribution in a system; closer to heavier components.

Velocity of Center of Mass

vcm = Σ(mi vi)/Σ(mi).

Total Momentum and Center of Mass

The total momentum of a system equals total mass times the velocity of its center of mass: Ptotal = M*vcm.

Center of Mass Motion Principle

The system’s center of mass moves as if all mass were concentrated there and all external forces acted on it.

External Forces and CM Motion

The net external force equals total mass times the CM acceleration: Fext = M*acm.

Effect of Internal Forces

Internal forces cancel and do not affect the center of mass motion.

Projectile Explosion Concept

"After a projectile explodes

Tug-of-War on Ice Concept

"Two people pulling on a rope move toward each other

Key Idea: Momentum Conservation

"The total momentum of an isolated system remains constant

Key Idea: Impulse and Safety

"Increasing impact time reduces force

Key Idea: Elastic vs. Inelastic Collisions

"Elastic: momentum and energy conserved. Inelastic: only momentum conserved

Key Idea: Center of Mass and Forces

Only external forces can change the motion of a system’s center of mass; internal forces cannot.