Acceleration due to Gravity of the Earth

Every falling object near the Earth's surface accelerates downward at the same rate, regardless of its own mass -- this is a direct consequence of combining Newton's law of gravitation with his second law.

Derivation. The gravitational force Earth exerts on a nearby mass $m$ is $\vec{F}=-\dfrac{GmM_e}{R_e^2}\hat{r}$. Setting this equal to $m\vec{a}$ from Newton's second law and cancelling $m$ gives the acceleration due to gravity directly:

$\vec{a}=-\frac{GM_e}{R_e^2}\hat{r}, \qquad |\,g\,| = \frac{GM_e}{R_e^2}. \qquad (6.43,\,6.44)$ …