Newton's Laws of Motion

Sir Isaac Newton's three laws of motion form the foundation of classical mechanics.

First Law: Law of Inertia

An object at rest stays at rest, and an object in motion stays in motion with the same speed and direction unless acted upon by an unbalanced force.

Key concept: Inertia - the resistance of an object to change its state of motion.

Examples

• A book on a table remains stationary
• A hockey puck slides across ice until friction stops it
• Passengers lurch forward when a car brakes suddenly

Second Law: F = ma

The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

Formula:

$\vec{F} = m\vec{a}$

Or more precisely:

$\sum \vec{F} = m\vec{a}$

Where:

• $F$ = Force (Newtons, $\text{N}$)
• $m$ = mass (kilograms, $\text{kg}$)
• $a$ = acceleration ($\text{m/s}^2$)

Example Calculation

A $2\text{ kg}$ object experiences a net force of $10\text{ N}$. Find its acceleration:

$a = \frac{F}{m} = \frac{10\text{ N}}{2\text{ kg}} = 5\text{ m/s}^2$

Code Example: Force Calculator

def calculate_force(mass: float, acceleration: float) -> float:
    """
    Calculate force using F = ma
    
    Args:
        mass: Mass in kg
        acceleration: Acceleration in m/s^2
    
    Returns:
        Force in Newtons
    """
    return mass * acceleration
 
# Example
mass = 50  # kg
acceleration = 2  # m/s^2
force = calculate_force(mass, acceleration)
print(f"Force: {force} N")  # Output: Force: 100 N

Applications

• Heavier objects require more force to accelerate
• Same force produces greater acceleration in lighter objects
• Fundamental to calculating motion in engineering

Third Law: Action-Reaction

For every action, there is an equal and opposite reaction.

$\vec{F}_{12} = -\vec{F}_{21}$

Where $\vec{F}_{12}$ is the force on object 1 by object 2, and $\vec{F}_{21}$ is the force on object 2 by object 1.

Examples

• Rocket propulsion: gases pushed down with force $F$, rocket pushed up with force $-F$
• Swimming: push water backward, water pushes you forward
• Walking: push ground backward, ground pushes you forward

Problem-Solving Example

A $1000\text{ kg}$ car accelerates from rest to $20\text{ m/s}$ in $4\text{ s}$. Calculate:

1. Acceleration: $a = \frac{\Delta v}{\Delta t} = \frac{20 - 0}{4} = 5\text{ m/s}^2$

2. Net Force: $F = ma = 1000 \times 5 = 5000\text{ N}$

3. Distance traveled (using $s = ut + \frac{1}{2}at^2$): $s = 0 + \frac{1}{2}(5)(4^2) = 40\text{ m}$

Python Simulation

import numpy as np
import matplotlib.pyplot as plt
 
def simulate_motion(mass, force, time):
    """
    Simulate motion under constant force
    """
    acceleration = force / mass
    t = np.linspace(0, time, 100)
    velocity = acceleration * t
    position = 0.5 * acceleration * t**2
    
    return t, velocity, position
 
# Simulate a 10 kg object with 20 N force for 5 seconds
t, v, s = simulate_motion(mass=10, force=20, time=5)
 
print(f"Final velocity: {v[-1]:.2f} m/s")
print(f"Final position: {s[-1]:.2f} m")

Problem-Solving Steps

1. Draw a free body diagram
2. Identify all forces acting on the object
3. Apply $\sum \vec{F} = m\vec{a}$ in relevant directions
4. Solve for unknown variables

These laws are essential for understanding motion in everything from sports to space travel.