Free fall
Galileo’s odd number rule
[chatgpt]
Galileo’s law of odd numbers describes how the distances fallen by an object in free fall increase in proportion to the sequence of odd numbers over equal intervals of time. This observation helped Galileo understand the uniform acceleration of freely falling objects.
The Law of Odd Numbers in Free Fall:
When an object falls freely under gravity (assuming no air resistance), it accelerates uniformly. Galileo discovered that if we divide the fall time into equal intervals (say 1 second each), then:
- The distance fallen in each successive time interval follows the sequence of odd numbers (1, 3, 5, 7, …).
- Total distance fallen after each interval is proportional to the square of the time elapsed.
How It Works:
Let’s break down these ideas.
- Uniform Acceleration and Increasing Velocity:
- When an object falls freely, its velocity increases at a constant rate due to gravity (\( g \approx 9.8 \, \text{m/s}^2 \)).
- After 1 second, the object reaches a velocity of \( g \) (9.8 m/s), after 2 seconds, \( 2g \), and so on.
- When an object falls freely, its velocity increases at a constant rate due to gravity (\( g \approx 9.8 \, \text{m/s}^2 \)).
Distance Covered in Each Interval:
- In the first time interval, the object falls a distance proportional to \(1\).
- In the second time interval, it falls an additional distance proportional to \(3\).
- In the third time interval, it falls an additional distance proportional to \(5\), and so on.
These distances correspond to the odd numbers sequence.
- In the first time interval, the object falls a distance proportional to \(1\).
- Cumulative Distance and Time Squared Relationship:
- The total distance fallen after each time interval corresponds to the sum of the first few odd numbers.
- According to Galileo’s observations, the cumulative distance fallen after each interval is proportional to the square of the elapsed time.
- The total distance fallen after each time interval corresponds to the sum of the first few odd numbers.
Example:
Imagine the fall time is divided into seconds:
- In the first second: The object falls a distance proportional to \(1\) (first odd number).
- In the second second: The object falls an additional distance proportional to \(3\) (next odd number).
- In the third second: It falls an additional distance proportional to \(5\).
So, if we measure the total distances fallen by the end of each second:
- After 1 second: \( 1 \) (total distance proportional to \(1^2\)).
- After 2 seconds: \( 1 + 3 = 4 \) (total distance proportional to \(2^2\)).
- After 3 seconds: \( 1 + 3 + 5 = 9 \) (total distance proportional to \(3^2\)).
This pattern continues, demonstrating that the total distance fallen after each second is proportional to \( t^2 \), where \( t \) is the time elapsed. This insight was a crucial step for Galileo in understanding uniform acceleration and laid the groundwork for classical mechanics.