In a vacuum, all objects fall at the same rate regardless of their mass. This is because there is no air resistance in a vacuum, so the only force acting on the objects is gravity. The force of gravity is the same for all objects, regardless of their mass, so they all fall at the same rate.

This was famously demonstrated by Galileo Galilei in the 16th century. He dropped two balls of different masses from the top of the Leaning Tower of Pisa and showed that they hit the ground at the same time. This experiment proved that the rate of fall of an object is independent of its mass.

The equation for the acceleration due to gravity is:

$$a = g$$

Where:

- \(a\) is the acceleration due to gravity (in meters per second squared)

- \(g\) is the gravitational constant (approximately 9.8 meters per second squared)

As can be seen from this equation, the acceleration due to gravity is a constant. This means that all objects fall at the same rate, regardless of their mass.