Momentum and Collisions

Momentum & Collisions: The Secret Language of Moving Things 🎱

Imagine you’re watching a game of pool. The white ball smashes into a red ball, and—CRACK!—the red ball flies away while the white one slows down or stops. What just happened? Welcome to the magical world of momentum!

🚀 What is Linear Momentum?

The Big Idea: Momentum is how much “oomph” a moving object has.

Think of a shopping cart. An empty cart rolling slowly has a little oomph. But a cart full of groceries rolling fast? That has a LOT of oomph. That “oomph” is momentum!

The Magic Formula

Momentum = Mass × Velocity
p = m × v

• p = momentum (measured in kg·m/s)
• m = mass (how heavy something is, in kg)
• v = velocity (how fast AND which direction, in m/s)

Simple Example 🛒

A 2 kg toy car moving at 3 m/s has:

p = 2 kg × 3 m/s = 6 kg·m/s

Key Insight: Momentum cares about DIRECTION. A car going left and a car going right have opposite momenta, even at the same speed!

💥 What is Impulse?

The Big Idea: Impulse is a “push over time” that changes momentum.

Imagine catching an egg. If you catch it with stiff hands (short time), it breaks! But if you let your hands move back with the egg (longer time), it survives. Same force, different time = different result!

The Impulse Formula

Impulse = Force × Time
J = F × Δt

• J = impulse (measured in N·s)
• F = force (in Newtons)
• Δt = how long the force acts (in seconds)

Simple Example 🥚

Catching a ball:

• Force = 10 N
• Time = 0.5 seconds
• Impulse = 10 × 0.5 = 5 N·s

🔗 The Impulse-Momentum Theorem

The Big Idea: Impulse equals the change in momentum!

This is beautiful: the “push over time” (impulse) exactly equals how much the momentum changes.

J = Δp = p_final - p_initial
F × Δt = m × v_final - m × v_initial

Why Car Airbags Work 🚗💨

When you crash:

• Your momentum MUST change (from moving to stopped)
• Airbag increases the TIME of the collision
• Same momentum change, but LESS force on your body!

graph TD
    A[You're Moving Fast] --> B["Crash Happens!"]
    B --> C{With Airbag?}
    C -->|Yes| D["Long Time<br/>Small Force<br/>Safe!"]
    C -->|No| E["Short Time<br/>Big Force<br/>Ouch!"]

Simple Example 🎾

A 0.1 kg ball hits a wall at 5 m/s and bounces back at 5 m/s:

• Initial momentum: 0.1 × 5 = 0.5 kg·m/s
• Final momentum: 0.1 × (-5) = -0.5 kg·m/s
• Change in momentum: -0.5 - 0.5 = -1 kg·m/s

⚖️ The Momentum Conservation Law

The Big Idea: In a closed system, total momentum never changes!

This is one of nature’s most powerful rules. When two pool balls collide, momentum isn’t created or destroyed—it just transfers from one ball to another!

Total Momentum Before = Total Momentum After
m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'

When Does Conservation Work?

✅ YES: No outside forces (like friction or hands pushing) ❌ NO: When external forces act on the system

Simple Example 🛹

Two skaters push off each other:

• Before: Both at rest, total momentum = 0
• After: One goes left, one goes right
• Their momenta are equal and opposite: still adds to 0!

graph TD
    A["Two Skaters<br/>Standing Still<br/>p = 0"] --> B["They Push Off"]
    B --> C["Skater 1<br/>Goes Left<br/>p = -10"]
    B --> D["Skater 2<br/>Goes Right<br/>p = +10"]
    C --> E["Total = -10 + 10 = 0 ✓"]
    D --> E

💢 Collisions: When Objects Meet

The Big Idea: Collisions are when two objects interact briefly and exchange momentum.

Every collision follows momentum conservation. But what makes collisions different is what happens to ENERGY.

Two Big Questions in Every Collision:

1. Is momentum conserved? ALWAYS YES (if no external forces)
2. Is kinetic energy conserved? Depends on the collision type!

🎭 Types of Collisions

1. Elastic Collision (Perfect Bounce) 🎱

What happens: Objects bounce off perfectly. Both momentum AND kinetic energy are conserved!

Real examples:

• Pool balls hitting each other
• Air molecules bouncing around
• Ideal physics problems 😄

Momentum: m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'
Energy: ½m₁v₁² + ½m₂v₂² = ½m₁v₁'² + ½m₂v₂'²

2. Inelastic Collision (Some Energy Lost) 🚗

What happens: Objects collide and some kinetic energy turns into heat, sound, or deformation.

Real examples:

• Cars crashing (crumple zones absorb energy)
• Dropping a ball (doesn’t bounce back to same height)
• Most real-world collisions!

3. Perfectly Inelastic Collision (Maximum Sticking) 🤝

What happens: Objects stick together after collision. Maximum kinetic energy is lost!

Real examples:

• Two clay balls hitting and sticking
• Football tackle where players move together
• Catching a ball

m₁v₁ + m₂v₂ = (m₁ + m₂)v_final

Simple Example: Catching a Ball ⚾

• Ball (0.2 kg) moving at 10 m/s toward you
• You (50 kg) standing still
• After catch, you both move together

Before: 0.2 × 10 + 50 × 0 = 2 kg·m/s After: (0.2 + 50) × v_final = 2 kg·m/s v_final = 2 ÷ 50.2 ≈ 0.04 m/s (barely moving!)

🔢 Coefficient of Restitution (e)

The Big Idea: A number that tells you how “bouncy” a collision is!

e = (Speed of Separation) ÷ (Speed of Approach)
e = (v₂' - v₁') ÷ (v₁ - v₂)

The Bounce Scale:

Simple Example 🏀

Drop a basketball from 2 meters. It bounces back to 1.5 meters.

• Speed down: √(2 × 10 × 2) ≈ 6.3 m/s
• Speed up: √(2 × 10 × 1.5) ≈ 5.5 m/s
• e = 5.5 ÷ 6.3 ≈ 0.87 (pretty bouncy!)

graph TD
    A["e = 1.0<br/>Perfect Bounce<br/>🎱"] --> B["e = 0.87<br/>Basketball<br/>🏀"]
    B --> C["e = 0.60<br/>Tennis Ball<br/>🎾"]
    C --> D["e = 0.30<br/>Soft Ball<br/>🥎"]
    D --> E["e = 0<br/>Clay Blob<br/>💩"]

📐 Collision Geometry: Direction Matters!

The Big Idea: Collisions happen in 2D and 3D, not just on a straight line!

Head-On (1D) Collision

Objects moving on the same line, toward each other.

• Easiest to calculate
• Only consider one direction

Oblique (2D) Collision

Objects hit at an angle. Momentum must be conserved in BOTH x and y directions!

x-direction: m₁v₁ₓ + m₂v₂ₓ = m₁v₁ₓ' + m₂v₂ₓ'
y-direction: m₁v₁ᵧ + m₂v₂ᵧ = m₁v₁ᵧ' + m₂v₂ᵧ'

Simple Example: Pool Shot 🎱

• Cue ball hits stationary ball at an angle
• After collision, balls go in different directions
• Total momentum in x-direction: conserved!
• Total momentum in y-direction: conserved!

graph TD
    A["Cue Ball Coming In<br/>at Angle"] --> B["IMPACT!"]
    B --> C["Ball 1<br/>Goes One Way"]
    B --> D["Ball 2<br/>Goes Another Way"]
    C --> E["Both X and Y<br/>Momentum Conserved!"]
    D --> E

The Angle Rule

In 2D elastic collisions with equal masses (like pool):

• If one ball is stationary before collision
• After collision, the balls go off at 90° to each other!

🌟 The Big Picture

graph TD
    A["MOMENTUM<br/>p = mv"] --> B["Changed By<br/>IMPULSE<br/>J = FΔt"]
    B --> C["CONSERVED<br/>In Collisions"]
    C --> D{Collision Type?}
    D --> E["Elastic<br/>e = 1<br/>Energy Kept"]
    D --> F["Inelastic<br/>0 < e < 1<br/>Some Lost"]
    D --> G["Perfectly<br/>Inelastic<br/>e = 0<br/>Stick Together"]

🎯 Key Takeaways

1. Momentum = Mass × Velocity — the “oomph” of motion
2. Impulse = Force × Time — the “push over time” that changes momentum
3. Impulse = Change in Momentum — the beautiful connection!
4. Momentum is ALWAYS conserved in isolated systems
5. Elastic collisions preserve kinetic energy; inelastic don’t
6. Coefficient of Restitution (e) measures bounciness (0 to 1)
7. 2D collisions require momentum conservation in ALL directions

Remember: Every time you watch billiard balls scatter, cars bump, or athletes collide, you’re watching momentum in action. The universe keeps perfect score—momentum is never created or destroyed, just shared! 🎱✨