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→Newtonian Gravitation: Formula with r^3
*''<b>r</b>'' is the vector that separates the objects' centers of mass.
*''<b>r</b>'' is the vector that separates the objects' centers of mass.
* <math>\mathbf{\hat{r}}</math> is the unit vector of ''<b>r</b>''.
* <math>\mathbf{\hat{r}}</math> is the unit vector of ''<b>r</b>''; <math>\mathbf{r} = \vert {\mathbf{r}} \vert \ \mathbf{\hat{r}}</math>
*''G'' is the [[gravitational constant]]: <math>\frac{6.67428 \times 10^{-11} N m^2}{kg^2}</math>
*''G'' is the [[gravitational constant]]: <math>\frac{6.67428 \times 10^{-11} N m^2}{kg^2}</math>
Another way to express this equation, that is more convenient for programming and expansion into power series, is:
:<math>\mathbf{F} = -G \frac{m_1 m_2}{\vert {\mathbf{r}} \vert ^3} \mathbf{r}</math>
Near the surface of the Earth, where the difference in gravitational acceleration at different heights is very small, Newtonian gravitation "reduces" to, or is mathematically equivalent to within a known margin of error, Galilean gravitation. That is, both Galilean gravitation and Newtonian gravitation can reasonably describe the motion of falling apples and artillery shells. But further from the surface of the Earth, the errors in Galileo's approximation become too great, and Newton's model is required.
Near the surface of the Earth, where the difference in gravitational acceleration at different heights is very small, Newtonian gravitation "reduces" to, or is mathematically equivalent to within a known margin of error, Galilean gravitation. That is, both Galilean gravitation and Newtonian gravitation can reasonably describe the motion of falling apples and artillery shells. But further from the surface of the Earth, the errors in Galileo's approximation become too great, and Newton's model is required.