Drag Equation Calculator

The drag equation calculator helps you compute a force exerted on a moving object immersed in a fluid. Reading the text below, you will discover what the equation for drag force is, how the shape of the object influences the force and what the drag coefficient is.

If an object moves through a fluid, it experiences a resisting force, the drag force. The value of this force depends on the size and shape of an object, the density of the fluid, and the relative velocity of the object and the fluid. The equation is:

Fd = 1/2 × ρ × u² × A × Cd

where:

• Fd – Drag force;
• ρ – Liquid's density;
• u – Relative velocity;
• A – Reference area; and
• Cd – Drag coefficient.

The reference area A for an object of a simple shape is the cross-sectional area orthogonal to the direction of the motion. For example, for a sphere of radius r, we would simply take A = π × r². For an object of a more complicated shape, like a car, the reference area is more difficult to specify, but usually, it's larger than the cross-sectional area.

For an application of the drag force, you can check the free fall with air resistance calculator and the Stokes' law calculator.

The drag force equation depends on the drag coefficient Cd. What is this? The drag coefficient is a dimensionless number that depends on the shape of the object. If an object has a smooth shape, then Cd is a small number, and the resulting drag force is small as well. For example, it's equal to 0.04 for a streamlined body, whereas it equals 1.05 for a cube.

The drag coefficient itself depends on the Reynolds number Re. The dependence is mild for a small enough Reynolds number (of a few thousand or smaller), and the drag coefficient is approximately constant. Check the Reynolds number calculator to learn more about different flows and their classification with Reynolds number.

How do you use our drag equation calculator? Simply choose a liquid, immersed object, and the relative velocity between them.

For example, for olive oil, the density ρ = 920 kg/m³, with an immersed long cylinder (Cd = 0.82) with cross-section A = 1 cm² and the relative velocity u = 3 m/s the resulting drag force is Fd = 0.3395 N.

You can compute the drag coefficient using the drag force equation. To do so, perform the following steps:

1. Take the fluid density where the object is moving.

2. Multiply it by the reference cross-sectional area and by the square of the relative velocity of your object.

3. Find the value of the drag force over your object and multiply it by 2.

4. Divide the last by the result of step 2 to get your drag coefficient as a non-dimensional quantity.

The terminal velocity occurs when there is a zero net force between the gravitational force and the drag force, which means that:

F_d = m⋅g

F_d = 0.5⋅ϱ⋅v_T²⋅A⋅C_d

From these equations, we find that:

v_T = √ 2⋅m⋅g/(ϱ⋅A⋅C_d )

The zero net force means an object will fall at a constant speed when it reaches the terminal velocity.

The drag force can be applied to study the motion and the design of vehicles, rockets, and airplanes. Designers can use it to develop sports equipment for cycling, swimming, and skiing, for instance, improving the athletes' performance by reducing the drag.

Another application is architecture, where engineers investigate better ways to design a building or a bridge under the effect of drag forces.

Let us consider that u = 20 m/s is the velocity of a skydiver falling with an open parachute. If the air density is ϱ = 1.2041 kg/m³, the area of the open parachute is A = 7 m², and its typical drag coefficient is C_d = 1.3, we can use the equation:

F_d = 0.5⋅ϱ⋅u²⋅A⋅C_d

So, for this example, the drag force is:

F_d = 2191.5 N