Lift is created by deflecting a flow of
air, and drag is generated on a body in a
wide variety of ways. From Newton's Second
Law of Motion, the
aerodynamic forces
on the body (lift and
drag) are directly related to the change in momentum of the fluid
with time. The **fluid momentum** is equal to the mass (m)times the
velocity (V) of the fluid.
Since the air moves, defining the mass is tricky. If the mass of
fluid were brought to a halt, it would occupy some volume in space.
And we could define its density (r) to be the
mass divided by the volume. But since the fluid is moving, we must
consider, instead, the mass flow rate, which
is the amount of mass passing a given point during some **time**
(mass/time). We can relate the mass flow rate
to the density mathematically.
The mass flow rate (mass/time) is equal to the density (mass/volume)
times the velocity (length/time) times the area.
With knowledge of the mass flow rate, we can express the aerodynamic
force as equal to the mass flow rate times the velocity. A quick units
check shows that the force (mass times the velocity divided by the time) is
equal to the mass flow rate (mass divided by time) times the velocity.
Therefore, the aerodynamic force is equal to a constant times
the density times the velocity
times the area times the velocity. Combining the velocity dependence and absorbing
the area into the constant, we find that the aerodynamic force is equal to a
constant times the density times the velocity squared.

__Effect of Velocity on Aerodynamic Forces __

The velocity used in the aerodynamic equation is the relative velocity between an object and the flow. The aerodynamic force depends on the square of the velocity. Doubling the velocity quadruples the force.

__Effect of Air Density on Aerodynamic Forces__

The aerodynamic force depends linearly on the density
of the air. Halving the density halves the force. As altitude
increases, the air density decreases. This explains why airplanes
have a **flight ceiling**, an altitude above which it cannot fly.
As an airplane ascends, a point is reached where there is not enough
air mass to generate enough lift to overcome the airplane's weight.
The relation between altitude and density
is a fairly complex exponential.

__FoilSim Applet__

You can investigate the effect of momentum on lift by using the FoilSim II Java Applet. Set a small angle of attack using the slider, then vary the "Speed" and "Altitude." Try doubling the speed and notice the effect on lift. Change the altitude until the air density is half of its previous value. What happened to the lift? You can use the browser "Back" button to return to this page. If your browser does not support JAVA, or you just want your own copy of FoilSim to play with, you can download it at no charge.

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- Beginner's Guide Home Page

*byTom
Benson
Please send suggestions/corrections to: benson@grc.nasa.gov *