# Differential Equations : Free Fall and Terminal Velocity

An object in free fall in a gravitational field is governed by the ODE m*dv/dt=mg + Fs, where m is the mass of the object, g=9.8 meters/sec is the acceleration of gravity, v(t) is the velocity of the object t seconds after it is released, and Fs denotes external forces acting on the object. In all that follows, assume that v(0)=0. In this problem, since we will investigate free fall and terminal velocity, let's choose the positive direction for velocity and position as downward in the same direction as g; therefore, the coefficient of mg in the ODE is +1, not -1.

1. If there are no external forces acting on the object, then its velocity increases without bound (until the object collides with something). This is unrealistic for motion in the earth's atmosphere, since air resistance is a significant effect. Therefore, assume that air resistance is present and is described by Fs=-kv (k is a constant and the minus sign indicates that the air resistance opposes motion). Show (include all steps) that the solution of this ODE is v(t)=mg/k(1-e^(kt/m)).

2. What is the terminal velocity vT=lim as t approaches infinity of a 100-kilogram object (a small linebacker or a large flower pot) subject to air resistance described by k = 5kg/sec?

3. Find the function that describes the position x(t) of the object for all t greater than or equal to 0 assuming that x=0 corresponds to the position at which the object is dropped.

4. Make a rough sketches of v(t) and x(t).

5. Now assume that Fa=-kv^2 (this is generally a more accurate way to model air resistance). Solve the resulting ODE for the velocity of the object.

6. With values of m=100kg and k=.1kg/meter, what is the terminal velocity of the object? (notice that the k's that appear in the two models are different.)

7. Find the function that describes the position x(t) of the object for all assuming that x=0 corresponds to the position at which the object is dropped.

8. Make rough sketches of v(t) and x(t).

9. In at least two paragraphs, compare and contrast the two models for air resistance.

#### Solution Summary

This well-presented solution shows how to solve differential equations related to free fall and terminal velocity. The response was given a rating of "5/5" by the student who originally posted the question.