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Differential Equations

The following is a quick example of a practical differential equations problem:

Free Falling Problem:
A ball of mass m is dropped with an initial velocity of v_0 from a plane. Assume the gravitation force is constant and that the force due to air resistance is proportional to the velocity of the ball. The following calculations model the ball's speed and distance fallen and are plotted below.

Given data:
m = 75, g = 9.81, k = 15

Initial conditions:
t_0 = 0, v_0 = 0

From Newton's second law:
where

Integrating to find the velocity as a function of time yields:

As a result of the air resistance, the velocity of the ball will reach a maximum:

Integrating the function again and using the initial condition yields an equation for the distance traveled as a function of time: