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

It is usual to use $$g$$ for the quantity $$\frac {G M} {r^2}$$.

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