2D Objects with Drag

Objective

• Find the equations that describe the path of an object with drag force.

Solution

Step 1: Describe the General 1D Case

In 1D, the forces of a vertically thrown object can be described as:

Since:

where:

• is the coefficient of drag,
• is velocity as a function of time

and:

Substituting it gives us:

Since:

We express sum of forces as

Then, we can now express this as:

Solving the differential equation in terms of allows us to find an equation of motion.

Step 2: Consider Forces in 2D

The model from before works only in 1D.
If we consider forces in 2D, some similarities may arise.

In 2D, gravity only works vertically.
However, drag opposes the force of motion, and therefore has a horizontal and vertical component.

Therefore:

We then break down the components of drag using basic trigonometry:

Substituting them, and considering that :

Since :

Step 3: Solving the Differential Equation in the x-case.

Isolate in one side.

Separate the variables and group related terms.

Integrate both sides, then solve for .

Solve for the initial value problem .

Since , we substitute it to the original problem.

To get an equation for position, we consider that .

We then solve the initial value problem .

Substituting it to the previous equation yields the position equation for on object with drag in the -direction.

Step 4: Solve the Differential Equation in the y-case.

Isolate in one side.

Rewrite in standard form of a linear differential equation.

We identify the coefficients of the linear differential equation.

Now, we find the integrating factor .

Multiplying the entire equation by the integrating factor, we get:

Since :

Now, solve for the initial value problem: .

Therefore:

Since :

Solve the initial value problem .

Therefore: