Sure! To explain the integration of Galileo's free fall in a simple way, let's first break down the concepts without getting too technical.

1. Galileo's Free Fall: A Quick Overview

Galileo discovered that all objects, regardless of their mass, fall to the ground at the same rate when there is no air resistance. This rate of acceleration is called gravitational acceleration and it is typically denoted as g, which is approximately 9.8 m/s² on Earth. This means that for every second an object falls, its speed increases by 9.8 meters per second.

2. The Equation of Free Fall (with constant acceleration)

The motion of an object in free fall can be described using some basic physics formulas. The key one is: $v(t) = g \cdot t$ Where:

• $v(t)$ is the velocity (speed) of the object after time $t$.
• $g$ is the acceleration due to gravity (9.8 m/s²).
• $t$ is the time the object has been falling.

Now, here's the important part: this equation tells us how the speed changes over time, but it doesn't tell us how far the object has fallen. To find the distance the object has fallen, we need to integrate this equation.

3. What is Integration?

You can think of integration as a way to calculate the total amount of something when you know how it changes over time. In the case of free fall, you are trying to find how far the object has fallen after a certain amount of time, given that its speed is constantly increasing.

For example, if an object is speeding up constantly (as it would in free fall), you can't just multiply speed by time to get distance because the speed is changing. Instead, you need to "add up" all the small changes in distance over time, and that's where integration comes in.

4. How Do We Integrate the Velocity Equation?

The velocity $v(t)$ tells you how fast the object is moving at any given time. To find the distance the object has fallen (which is the total distance traveled), you need to "sum up" the velocity at each moment in time.

In mathematical terms, this means integrating the velocity function $v(t) = g \cdot t$. The result of this integration gives you the equation for the distance fallen $d(t)$: $d(t) = \int v(t) \, dt = \int g \cdot t \, dt$ When you do this integration, you get: $d(t) = \frac{1}{2} g \cdot t^2$ This tells you how far the object has fallen after time $t$.

5. The Meaning of the Result

The equation $d(t) = \frac{1}{2} g \cdot t^2$ means that the distance fallen increases with the square of the time. So, if you let the object fall for 1 second, it falls a certain distance; if you let it fall for 2 seconds, it falls four times as far, and so on.

Recap

• Galileo showed that objects fall with constant acceleration (g).
• We use integration to find how far an object has fallen over time when its velocity is changing.
• The formula for the distance fallen is $d(t) = \frac{1}{2} g \cdot t^2$.

Would you like a more detailed breakdown or to explore another example? Feel free to ask any questions!

Here are some related questions to explore further:

1. How does air resistance affect free fall?
2. What happens if the object is thrown upwards instead of dropped?
3. How can we calculate the velocity at any given time using integration?
4. How does the concept of constant acceleration apply to other forces in physics?
5. How is this equation used in real-world situations like dropping an object from a height?

Tip: When learning integration, it's helpful to think of it as "adding up" tiny pieces of change over time. It’s like measuring the total distance traveled by adding up how far you’ve gone each moment!