Exercise 7

When a skydiver jumps from a plane, gravity causes her downward velocity to increase at the rate of $g\approx 9.8$ meters per second squared. At the same time, wind resistance causes her velocity to decrease at a rate proportional to the velocity.

1. Using $k$ to represent the constant of proportionality, write a differential equation that describes the rate of change of the skydiver's velocity.

2. Find any equilibrium solutions and decide whether they are stable or unstable. Your result should depend on $k$.

3. Suppose that the initial velocity is zero. Find the velocity $v(t)$.

4. A typical terminal velocity for a skydiver falling face down is 54 meters per second. What is the value of $k$ for this skydiver?

5. How long does it take to reach 50% of the terminal velocity?