# How to account for air resistance?

• I
All the physics classes ive ever had modled object motion as though it where in a vaccuum, becuase air resistance is obviously a pain in the but to account for.

Mathmatically how would i account for it?

Say after one second of free fall how much of my velocity would be lost due to air?

Im assuming it has to do with the ratio of mass of the object to its crossection to the air

## Answers and Replies

Buzz Bloom
Gold Member
Hi jonatron:

I suggest you use a browser and search on "air resistance formula".

Regards,
Buzz

FactChecker
Science Advisor
Gold Member
The effect of air resistance would be in the form of a force, called drag, that opposes the motion. Drag does not care about the mass of the object, only the shape and surface. It is proportional to velocity squared, air density, the cross sectional area of the object and a coefficient of drag that represents the shape of the object. See the first equation in https://en.wikipedia.org/wiki/Drag_(physics).

If the fall is short, you can assume that the air density is a constant. In general, the drag force must be calculated frequently for the changing air density and any rotation of the object that presents a different cross section. Then the motion of the object can be calculated for that force till the next drag calculation is done.

boneh3ad
Science Advisor
Gold Member
Drag does not care about the mass of the object, only the shape and surface.

The drag force itself does not, but its effect on the motion of the object certainly depends on the object's mass given that ##a = F/m##.

It is proportional to velocity squared, air density, the cross sectional area of the object and a coefficient of drag that represents the shape of the object. See the first equation in https://en.wikipedia.org/wiki/Drag_(physics).

I'd also like to point out (for the sake of the OP) that drag coefficients can be incredibly useful, but are ultimately just empirical approximations. Drag is actually an extraordinarily complicated quantity that, for many shapes (e.g. a sphere), can be estimated quite readily and accurately with a drag coefficient. More complex shapes (e.g. an airplane) are more difficult to handle.

If the fall is short, you can assume that the air density is a constant. In general, the drag force must be calculated frequently for the changing air density and any rotation of the object that presents a different cross section. Then the motion of the object can be calculated for that force till the next drag calculation is done.

Alternatively, just set the whole thing up as a differential equation and solve it to get a complete time history of the position, velocity, and drag for a given object. This can often be done without too many unreasonable assumptions for simple situations like a falling ball.

• FactChecker
The resistive force, or drag, f of the air is the force by the medium that oppose an object in its opposite direction of motion, given by:
$$f=f(v)\hat { v }$$
The function f(v) varies with v in a very complicated way. However, at lower speed(lower than the speed of sound), it is a good approximation to write
$$f(v)=bv+c{ v }^{ 2 }$$
The linear term arises from the viscous drag of the medium and is generally proportional to the viscosity of the medium and the linear size of the object. The quadratic term is proportional to the density of the medium and the cross-sectional area of the object.

For big and fast objects, the quadratic resistance dominates, for small and slow objects, linear resistance dominates.

Also for small objects, the drag coefficient may go down significantly as speed increases because the Reynolds number goes up.