Law of Conservation of Momentum

📚 Key Concepts

🔹 Conservation of Momentum

Conservation of Momentum: The total momentum of an isolated system remains constant if no external force acts on it.

Mathematical Statement:
Total momentum before collision = Total momentum after collision
p₁ + p₂ = p₁’ + p₂’

🔹 Conditions for Conservation

Requirements:

Isolated system: No external forces acting
Internal forces only: Forces between objects in the system
Newton’s Third Law: Action-reaction pairs are internal

Examples of Isolated Systems:

Two colliding balls on a smooth surface
Gun and bullet system during firing
Rocket and expelled gases in space

rocket

🔹 Types of Collisions

1. Elastic Collision:

Momentum and kinetic energy are conserved
Objects bounce back after collision
Example: Collision between two steel balls

2. Inelastic Collision:

Only momentum conserved, kinetic energy not conserved
Objects may stick or deform
Example: Car crash, clay balls sticking together

3. Perfectly Inelastic Collision:

Objects stick together after collision
Maximum loss of kinetic energy
Example: Bullet embedding in a wood block

collision

🔹 Real-life Applications

Gun Recoil:

Initially: Gun and bullet at rest (Total momentum = 0)
After firing: Bullet forward, gun backward
Formula: m₁v₁ + m₂v₂ = 0

Rocket Propulsion:

Gases expelled downward at high speed
Rocket gains upward momentum
Works even in vacuum due to conservation of momentum

Walking and Swimming:

Person pushes ground or water backward
Body moves forward with equal momentum

pool

🔹 Collision Problems

Head-on Collision:

Two objects move toward each other
After collision: velocities change but total momentum same
Formula: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Explosion Problems:

Initially at rest; after explosion, parts move in opposite directions
Total momentum remains zero

Basic Sign Convention Rules:

Choose a positive direction (e.g., right or forward)
Opposite directions are taken as negative

🔹 Importance in Physics

Fundamental Principle:

One of the core conservation laws in physics
Applies from atoms to galaxies
Predicts motion outcomes in collisions and explosions

explosion

Engineering Applications:

Vehicle safety (airbags, crumple zones)
Sports gear and design
Spacecraft propulsion and maneuvering
Industrial machine analysis (impact, vibration)

🧪 Important Formulas

Conservation of Momentum: p₁ + p₂ = p₁’ + p₂’
Gun Recoil: m₁v₁ + m₂v₂ = 0

🔹 Numerical Applications

Solution:

Initial momentum = 0
Final momentum = 4 × v + 0.02 × 400 = 0
Recoil velocity = -2 m/s (backward)

Solution:

Initial momentum = 2 × 5 + 3 × 2 = 16 kg⋅m/s
Final momentum = (2 + 3) × v = 5v
Final velocity = 16/5 = 3.2 m/s

Solution:

Taking right as positive direction: Initial: m₁ = 5 kg, u₁ = +8 m/s; m₂ = 3 kg, u₂ = -6 m/s Final: m₁ = 5 kg, v₁ = -2 m/s; m₂ = 3 kg, v₂ = ?
Conservation of momentum: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ 5(8) + 3(-6) = 5(-2) + 3v₂ 40 – 18 = -10 + 3v₂ 22 = -10 + 3v₂ 3v₂ = 32 v₂ = 10.67 m/s (to the right)

Solution:

Taking car’s direction as positive: Initial: m₁ = 1000 kg, u₁ = +20 m/s; m₂ = 1500 kg, u₂ = -15 m/s Final: Combined mass = 2500 kg, v = ?

Conservation of momentum: 1000(20) + 1500(-15) = 2500v 20000 – 22500 = 2500v -2500 = 2500v v = -1 m/s

The vehicles move at 1 m/s in the truck’s original direction.