Question 557922
Good question. Start with the general equation for vertical motion:


*[tex \LARGE x = -\frac{1}{2}at^2 + v_0t + x_0] where a is the acceleration due to gravity.


This equation can be derived using integral calculus (I'll prove it next). The acceleration due to gravity can be derived from the equation


*[tex \LARGE F = m_1a = G\frac{m_1m_2}{R^2}] where G is the gravitation constant, m1 and m2 are the masses of some object and Earth respectively, and R is the radius of the Earth. Plugging in, we find that a is approximately -9.8 m/s^2 (about -32 ft/s^2). Half of that is -16 ft/s^2, which is where the -16 comes from.


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To derive the vertical motion formula, start with the fact that


*[tex \LARGE v = \int a\, dt = at + v_0] (the v0 is our constant of integration)


Integrating again,


*[tex \LARGE x = \int v\, dt = \int at + v_0\, dt = \frac{1}{2}at^2 + v_0t + x_0]


And we are done.