Objects do fall at the same rate regardless of their mass. This was the conclusion of one of Galileo Galilei's numerous experiments. This fact was also theorized by Aristotle, but Galileo set up a controlled experiment to test this theory. The famous story goes that Galileo dropped balls of different weights from the top of the Leaning Tower of Pisa. Although there is no concrete proof that he actually performed this experiment, he did replicate this event with balls rolled down a ramp. He found that the balls rolled the same distance in the same time, regardless of the weight of the balls. This is because mass does not affect an object's acceleration due to gravity. This idea is embodied by Newton's second law of motion:

$$ F = ma $$

In this equation, \( F \) is force, \( m \) is mass, and \( a \) is acceleration. The acceleration due to gravity is about \( 9.8 m/s^2 \) near the surface of the Earth, and is denoted as \( g \). If we consider the force due to gravity, we get the following:

$$ F_g = mg$$

So the acceleration due to gravity is:

$$ a = \frac{F_g}{m} = \frac{mg}{m} = g$$

This equation shows that the acceleration due to gravity is independent of mass, which means objects with different masses should accelerate at the same rate.