Why do objects in a vacuum fall at the same rate?

Objects in a vacuum fall at the same rate due to the uniform acceleration caused by gravity, irrespective of their mass.

In more detail, this phenomenon is a direct consequence of the principle of equivalence, a cornerstone of Einstein's theory of general relativity. According to this principle, the effect of gravity on an object is indistinguishable from the effect of an object being in an accelerated frame of reference. This means that all objects, regardless of their mass, will experience the same acceleration when in a gravitational field, such as that of the Earth.

In a vacuum, there are no other forces acting on the objects, such as air resistance, which could otherwise affect their rate of fall. Therefore, the only force acting on the objects is gravity. The acceleration due to gravity near the surface of the Earth is approximately 9.81 m/s². This value is often rounded to 10 m/s² for simplicity in calculations.

This concept was famously demonstrated by Galileo, who is said to have dropped two spheres of different masses from the Leaning Tower of Pisa to show that they hit the ground at the same time. While the story is likely apocryphal, it illustrates the principle well. Galileo's experiments showed that the acceleration of an object in free fall is independent of its mass.

This principle is also demonstrated by astronauts in space. When they let go of objects of different masses, they observe that all objects fall towards the Earth at the same rate, regardless of their mass. This is because in the vacuum of space, there is no air resistance to slow down the objects.

In conclusion, the reason why objects in a vacuum fall at the same rate is due to the principle of equivalence and the absence of other forces such as air resistance. This results in a uniform acceleration caused by gravity, which is independent of the mass of the objects.