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.

If the force due to the gravitational field is the only force on the body, it is said to be in free fall.

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:
 * $F = G \dfrac {M m} {r^2}$

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:
 * $F = m \dfrac {G M} {r^2}$

But from Newton's Second Law of Motion:
 * $F = m a$

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

Hence:
 * $a = \dfrac {G M} {r^2}$

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