The equation relating acceleration, distance traveled, and time of fall is derived from the equations of motion for uniformly accelerated motion. Here's the breakdown:

Equation:

d = v₀t + (1/2)at²

Where:

* d is the distance traveled (also known as displacement)

* v₀ is the initial velocity (usually 0 for objects falling from rest)

* t is the time of fall

* a is the acceleration due to gravity (approximately 9.8 m/s² on Earth)

Explanation:

* The first term, v₀t, represents the distance traveled if the object were moving at a constant velocity.

* The second term, (1/2)at², accounts for the increasing velocity due to the acceleration of gravity.

Important Note: This equation assumes that the object is falling in a vacuum, meaning there's no air resistance. In reality, air resistance will affect the object's motion, causing it to fall slower.

Example:

Let's say an object falls from rest for 3 seconds. To find the distance traveled, we can plug the values into the equation:

d = (0)(3) + (1/2)(9.8)(3)²

d = 0 + 4.9 * 9

d = 44.1 meters

Therefore, the object would fall approximately 44.1 meters in 3 seconds.