By: Tao Steven Zheng (鄭濤)

【Problem】

Consider a skydiver falling out of an airplane. He opens his parachute during the descent. Approximate the external force on the skydiver, given that air resistance is proportional to the velocity squared .

Part 1: Use the force equation to calculate the terminal velocity .

Part 2: Change your force equation into a first-order differential equation. Then use your result to arrive at the integral

Part 3: Solve the above integral. Then rearrange the velocity solution as a function of time.

Hint

【Solution】

Part 1
There are two forces on the skydiver: the force of gravity and the force of air-resistance (drag force).

Since and , we get

The falling object is in equilibrium when it reaches terminal velocity, therefore,

Part 2
Newton's second law of motion states . Since , we get

Since ,

Use separation of variables to arrive at the integral

Part 3
Integrate using partial fractions or using the hint gives

Rearrange the above equation