When three different weights fall from a certain height under vacuum, it's a fundamental concept in physics that all objects, regardless of their mass or weight, will accelerate at the same rate and therefore take the same amount of time to reach the earth.

Falling Objects in a Vacuum

The key to understanding this phenomenon is the presence of a vacuum. A vacuum is a space devoid of matter, meaning there is no air or other gases present. On Earth, objects falling through the atmosphere experience air resistance, which is a force that opposes their motion. This air resistance depends on factors like the object's shape, size, and speed, and its presence causes lighter or less aerodynamic objects to fall slower than heavier or more streamlined ones.

However, in a vacuum, there is no air resistance to impede the fall. The only significant force acting on the falling weights is gravity. The acceleration due to gravity (\(g\)) near the Earth's surface is approximately \(9.8 \text{ m/s}^2\). This acceleration is constant for all objects, irrespective of their mass or weight. The formula for the distance an object falls under constant acceleration from rest is given by:

\(s = ut + \frac{1}{2}gt^2\)

Where:

• \(s\) is the distance (height from which the weights fall)
• \(u\) is the initial velocity (which is \(0\) since they fall from rest)
• \(g\) is the acceleration due to gravity
• \(t\) is the time taken

Since \(u = 0\), the equation simplifies to:

\(s = \frac{1}{2}gt^2\)

From this equation, we can see that if the height \(s\) and the acceleration due to gravity \(g\) are constant, then the time \(t\) taken to fall will also be constant. The mass or weight of the object does not appear in this equation.

Gravity's Consistent Pull

This principle was famously demonstrated by Galileo Galilei, who showed that in the absence of air resistance, a heavy object and a light object (like a feather and a hammer) would fall at the same rate. This concept is often illustrated by experiments performed on the Moon (where there's virtually no atmosphere), where a feather and a hammer dropped simultaneously reached the surface at the same time.

Therefore, when three different weights are dropped from the same height in a vacuum, they all experience the same gravitational acceleration and encounter no opposing forces. This results in them reaching the earth at precisely the same time.