The purpose of this page is to take a more mathematical look at air (fluid) resistance (also called drag or the drag force) and terminal velocity. Previously, we saw that the air resistance force on an object depends primarily on

- the relative velocity of the object and the fluid
- the shape of the object
- the density of the fluid
- other properties of the object, such as surface texture, as well as other properties of the fluid, such as viscosity

To keep our discussion under control, we will restrict our discussion to air resistance forces proportional to v and v^{2}.

Many sources attempt to treat all of the non-velocity influences on the drag force separately. So, we have a constant/variable that represents the influence of the shape of the object on the air resistance (drag) force, a constant/variable that represents effect of the density of the fluid on the air resistance force, etc. This results in a very large, very complicated looking expression for the air resistance force. If your life isn't complicated enough, I recommend that you switch to one of these treatments. My choice (and the choice of many others, too - it's not my idea) is to lump all of these other factors into one constant - let's call it "b". So, the shape of the object influences the value of "b", the density of the fluid influences "b", and so on.

This means that we can write the air resistance force (or drag force) as f_{drag} = bv for very small, slow objects, or f_{drag} = bv^{2} for "human-size objects, depending on the situation.

One straight-forward result of having a mathematical expression for the drag force is that we can easily write an expression for an object's terminal velocity. At terminal velocity, the drag force equals the weight, mg. of the object. If the drag force is proportional to velocity, then, when the velocity equals terminal velocity, we can write:

bv

_{term}= mg

where v_{term} is the terminal velocity. This means that:

In practice, it is easier (and more precise) to measure or estimate the terminal velocity of an object than to calculate the coefficient b. So this expression may be more useful in practice written as:

If f_{drag} = bv^{2}, the expression for terminal velocity becomes:

so:

What are the dimensions of "b" in each expression for f_{drag}?

(a) If f_{drag} = bv_{term}, then . Dimensions of b are . Therefore, this b would have SI units of "kg/s".

(b) If f_{drag} = bv^{2}, then . Now, dimensions of b are . So, in this case, b would have SI units of "kg/m".

A tiny particle of mass 4 x 10^{-4} kg (so f_{drag} = bv) has a drag coefficient, b = 3.3 x 10^{-2} kg/s. What is this particle's terminal velocity?

At terminal velocity,

bv

_{term}= mg

so:

A skydiver of mass 50 kg (f_{drag} = bv^{2}) has a terminal velocity of 60 m/s. What is the drag coefficient, b, for this skydiver?

At terminal velocity,

so:

*last update January 25, 2008 by JL Stanbrough*