**Problem**

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.

**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

- .