Our models for velocity of a falling object will all be based on Newton's Second Law of Motion, which states that force equals mass times acceleration:
Here F is the force exerted on an object of mass m, causing the object to have an acceleration a.
Acceleration is defined to be the derivative of velocity, i.e.,
where v = v(t) is the velocity at time t. Thus Newton's Law can be rewritten as
The primary force on a falling body is gravity, the pull of the Earth's mass on the object. Our first model for a falling body will consider gravity to be the only force on the object.
It is known through experimental observation that (near the surface of the Earth) the force of gravity on an object is proportional to the mass of the object, i.e., there is a constant g such that
The value of the constant g is known by experimentation to be approximately 32.2 sec2.
Equating our two formulas for the force F and dividing by m, we find a differential equation:
If we assume that our object was initially at rest at time t = 0, then our initial condition is
v(0) = 0. Together with the differential equation, we have an initial value problem for the velocity function v = v(t):
This solution for a velocity function leads to a second differential equation: The velocity v is itself the derivative of the distance function s = s(t), i.e.,