Consider a ball dropped form rest. The ball is subject to a drag force proportional to its velocity squared: .
Part 1: Determine the terminal velocity.
The terminal velocity is the velocity in which the speed no longer accelerates whilst falling.
Part 2: Determine the velocity over time.
Part 3: Determine the trajectory over time.
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.
To solve for the velocity over time, it is advantageous to transform the differential equation as follows. Remember the ball was originally at rest.
and the solution is
Solving for the velocity yields
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