Why do all objects fall at the same rate in a vacuum, independent of mass?

This is only the case in a vacuum because there are no air particles, so there is no air resistance; gravity is the only force acting. You can see it for yourselves with this easy experiment:

Take one piece of A4 paper and scrunch it up into a ball. Take two pieces of identical A4 paper and scrunch them up together into another ball. Your two paper balls should be of similar size but one twice as heavy as the other. Now drop them from the same height at the same time – you will see that they hit the ground at the same time! There is still air resistance but its effects are the same for both balls as they are the same size and shape. So it’s like there’s no air resistance at all!

Here are two different ways of explaining this phenomenon.

 

Explanation using equations:

Any object of mass m in a gravitational field (in this case Earth’s) has a gravitational force, F, acting on it:

F = (GmM) / R²

where G is the gravitational constant (this number does not change, it is the same throughout the whole universe), M is the mass of the Earth, and R  is the distance between the object and the centre of the Earth. It is this force which causes objects to fall to the ground in the first place.

Newton’s Second Law states that a force acting on an object will cause a change in speed, or acceleration, a, of the object:

F = ma        (Very important equation)

Therefore, the gravitational force will cause the object to accelerate towards the Earth. To find a formula for this acceleration, we combine the two equations for F above:

ma = (GmM) / R²

Then we can divide through by m to get:

a = (GM) / R²

As we can see, m does not appear in this formula, meaning that the acceleration of an object in free-fall does not depend on its mass.

 

“Wordy” explanation:

Gravity exerts a greater force on a heavy object than on a light object which is what you would expect. So why don’t heavy objects fall faster? The effect of this greater force on the acceleration of the object is cancelled out by the greater mass of the object. To help us understand this, let’s consider the following analogy. Imagine that you have to pull two boxes across a room; one box is twice as heavy as the other. In order to pull them at the same speed you need to pull the heavier box with twice as much force. Gravity pulling objects to the ground is like you pulling boxes across a room. Gravity needs to exert more force on heavier objects to make them fall as quickly as lighter objects.