Acceleration Due to Gravity

Physical Law
A body $$B$$ in a uniform gravitational field experiences a force which gives rise to a constant acceleration independent of the mass of the body.

Derivation
This law can be derived from Newton's Law of Universal Gravitation.

Let the mass of $$B$$ be $$m$$.

Let the mass of the body $$P$$ which gives rise to the gravitational field be $$M$$.

Then the force on $$B$$ is given by:
 * $$F = G \frac {M m} {r^2}$$

where:
 * $$G$$ is the gravitational constant;
 * $$r$$ is the distance between the centers of gravity of $$B$$ and $$P$$.

The assumption is that $$M$$ is orders of magnitude greater than $$m$$, and $$r$$ is also several orders of magnitude greater than the displacements observed on $$B$$ in the local frame.

Then we have:
 * $$F = m \frac {G M} {r^2}$$

But from Newton's Second Law, we have:
 * $$F = m a$$

where $$a$$ is the magnitude of the acceleration which would be imparted to the body if no other force were acting on it.

Hence we have:
 * $$a = \frac {G M} {r^2}$$

which is, as we said, independent of the mass of the object.

Gravity on Earth
When the body $$P$$ of mass $$M$$ is the Earth, and the $$B$$ of mass $$m$$ is located at or near its surface, it is usual to use $$g$$ for the quantity $$\frac {G M} {r^2}$$.

Therefore the force on $$B$$ can be expressed as:
 * $$F = m g$$

Historical Note
This result famously contradicts Aristotle, who taught that heavier objects fall faster than light ones.

It was Galileo who is supposed to have been the first one to determine the truth.

The fact that in the general case air resistance can not be ignored goes some way to explaining how the truth was not arrived at earlier. Even children notice how leaves, for example, flutter gently down from on high, whereas stones fall with alacrity.