Problem

Consider a ball dropped form rest. The ball is subject to a drag force proportional to its velocity squared: .

Drag force.jpg

Part 1: Determine the terminal velocity.

Hint

The terminal velocity is the velocity in which the speed no longer accelerates whilst falling.

Part 2: Determine the velocity over time.

Hint

Part 3: Determine the trajectory over time.

Hint:

Solution

Part 1

When the ball is released, the free-body diagram suggests that

.

The terminal velocity is the velocity in which the speed no longer accelerates whilst falling, so the net force must be zero.

Part 2

To solve for the velocity over time, it is advantageous to transform the differential equation as follows. Remember the ball was originally at rest.

Thus

and the solution is

.

Solving for the velocity yields

.

Part 3

To determine the trajectory, use the fact . Thus,

.

The right-hand-side is difficult to integrate; however, it can be expressed as

which makes it doable using integration by substitution. The result is

.
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