Dynamics

Dynamics - Complete Study Notes

🎯 Learning Objectives: Understand Newton's Laws, momentum, collisions, and motion with resistance. Apply these concepts to solve physics problems.

1. Newton's Laws of Motion

First Law (Law of Inertia)

Statement: Every object continues in its state of rest or uniform motion in a straight line unless acted upon by a resultant external force.

Key Insight: Force is needed to CHANGE motion, not maintain it.
Zero resultant force ⇒ Zero acceleration ⇒ Constant velocity

Example: Earth orbits at 30 km/s without pushing force. Space probes coast at constant velocity for years.

Second Law

Statement: The rate of change of momentum is proportional to the resultant force.

F = Δp/Δt

F = ma (for constant mass)

CRITICAL: Force (F) and acceleration (a) are VECTORS and always point in the SAME direction.

Definition: 1 Newton = force that gives 1kg mass an acceleration of 1 m/s²

Third Law (Action-Reaction)

Precise Statement: If body A exerts force on body B, then B exerts equal, opposite force on A of the SAME TYPE.

Action-Reaction Pairs:

Equal magnitude
Opposite direction
Same force type
Act on DIFFERENT bodies
Never cancel out!

2. Mass vs. Weight

Feature	Mass	Weight
Definition	Measure of inertia	Gravitational force
Nature	Scalar	Vector
SI Unit	Kilogram (kg)	Newton (N)
Variation	Constant everywhere	Depends on location

W = mg

Where: W = weight (N), m = mass (kg), g = gravitational field strength (N/kg)

Real-world Example: 5kg mass weighs ~49N on Earth, ~8N on Moon. Kicking it feels the same because MASS (inertia) is unchanged!

3. Linear Momentum

p = mv

Units: kg·m/s or N·s | Vector quantity - same direction as velocity

From Newton's Second Law: Change in momentum = Force × Time

4. Conservation of Momentum

Principle: Total momentum of isolated system remains constant (no external forces)

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

For 2D problems: Resolve into x and y components separately

5. Collisions & Energy

Elastic Collisions

Momentum CONSERVED
Kinetic Energy CONSERVED

(u₁ - u₂) = -(v₁ - v₂)

or (u₁ - u₂) = (v₂ - v₁)

Relative speed of approach = Relative speed of separation

Inelastic Collisions

Momentum CONSERVED
Kinetic Energy NOT conserved

m₁u₁ + m₂u₂ = (m₁ + m₂)v

Perfectly inelastic: objects stick together (maximum KE loss)

Important: Momentum ALWAYS conserved in isolated systems. Kinetic energy MAY OR MAY NOT be conserved.

6. Non-uniform Motion & Resistance

Friction & Drag Forces

Friction: Opposes motion between surfaces
Drag: Opposes motion through fluids (air/water)
Simple model: Drag force increases with speed

Terminal Velocity

Occurs when: Driving force = Resistive force
Result: Resultant force = 0 ⇒ Constant velocity

Example - Skydiver:
• Jump: weight >> air resistance ⇒ accelerate down
• Speed increases ⇒ air resistance increases
• Air resistance = weight ⇒ terminal velocity reached
• Parachute opens ⇒ larger area ⇒ larger drag ⇒ new terminal velocity

7. 2D Momentum Problems - Strategy

Resolve all velocities into x and y components
Apply momentum conservation to x-direction:
m₁u₁ₓ + m₂u₂ₓ = m₁v₁ₓ + m₂v₂ₓ
Apply momentum conservation to y-direction:
m₁u₁ᵧ + m₂u₂ᵧ = m₁v₁ᵧ + m₂v₂ᵧ
Combine results for final velocities

8. Problem-Solving Guide

Free-body Diagrams: Essential! Show all forces acting on ONE object.
Newton's 3rd Law: Check: Same type? Different objects?
Momentum: Define system → Check isolation → Assign direction → Apply conservation
Units: Always use kg, m/s, N for consistency

Progress Checklist

I can state and apply Newton's three laws

I understand mass vs. weight and can use W = mg

I can define momentum and use p = mv

I can solve momentum conservation problems in 1D and 2D

I can distinguish elastic vs. inelastic collisions

I understand terminal velocity and motion with resistance

I can use relative velocity equations for elastic collisions

I can draw and interpret free-body diagrams

🎯 Exam Tip: Practice identifying action-reaction pairs and drawing clear free-body diagrams - these are common exam topics!

Why Air Resistance (R) Increases with Velocity (v)

Air resistance is caused by collisions between the moving object and air molecules.

When an object moves slowly:

It collides with fewer air molecules per second, so the resisting force is small.

As the object speeds up:

It collides with more air molecules per second, and each collision happens with greater force.

This makes the upward resisting force (R) larger.

In short:

Low velocity → small R

Higher velocity → larger R

Terminal Velocity

Eventually R grows large enough to balance the weight (W).

R = W

At this point, the object reaches terminal velocity (constant speed, no acceleration).

Isolated System - Definition

Definition

An isolated system is a physical system that does not interact with its surroundings. No external forces act on the system, and there is no exchange of matter or energy with the environment.

🎯 Key Characteristics

No external forces - The net external force on the system is zero
No mass transfer - Matter cannot enter or leave the system
No energy transfer - Energy cannot enter or leave the system
Closed boundaries - The system is completely self-contained

Types of Systems

✅ Isolated System

No exchange of matter
No exchange of energy
No external forces
Momentum conserved
Energy conserved

❌ Non-Isolated System

Matter can enter/leave
Energy can enter/leave
External forces act
Momentum not conserved
Energy not conserved

Isolated System Diagram

Conservation Laws in Isolated Systems

Momentum Conservation

In an isolated system, the total momentum is always conserved:

Σp_initial = Σp_final

This is why we can use momentum conservation in collision problems when we define our system properly.

Energy Conservation

In an isolated system, the total energy is conserved (though it may change forms):

E_{total initial} = E_{total final}

Practical Examples

✅ Good Examples of Isolated Systems

System	Why Isolated	Conservation Applies
Two colliding balls in space	No gravity, no air resistance	Momentum conserved
Entire universe	Nothing outside to interact with	Energy and momentum conserved
Closed thermos flask	No heat or matter exchange	Energy conserved

❌ Non-Isolated Systems (Common Mistakes)

System	Why NOT Isolated	Conservation Fails
Ball falling to ground	Gravity (external force) acts	Momentum not conserved
Car collision on road	Friction with road surface	Momentum not conserved
Rocket launching	Expels mass, external forces	Momentum not conserved

Problem-Solving Strategy

How to Identify an Isolated System

Check for external forces - Are there forces from outside the system?
Check for mass exchange - Is matter entering or leaving?
Check for energy exchange - Is energy entering or leaving?
Define system boundaries - Make sure your system includes all interacting objects

Collision Example

Scenario: Two cars collide on an icy road (negligible friction)

System: Both cars together

Isolated? ✅ Yes - if we ignore air resistance and friction

Result: Momentum is conserved: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Common Exam Questions

Q: "Explain why momentum is conserved in the collision of two particles in deep space."

A: In deep space, there are no external forces (no gravity, no air resistance). The two particles form an isolated system, so the total momentum must be conserved according to Newton's laws.

Q: "A ball is dropped from height h. Why is momentum not conserved?"

A: Momentum is not conserved because gravity (an external force) acts on the ball. The ball alone does not form an isolated system. However, if we consider the ball + Earth as our system, then momentum is conserved.

🎯 Key Takeaway

An isolated system is crucial for applying conservation laws. Always check if your system is truly isolated before using momentum or energy conservation in calculations.

No external forces + No mass/energy exchange = ISOLATED SYSTEM