Aerodynamic forces
are generated whenever an object moves through a liquid or gas.
From Newton's second law
of motion, the aerodynamic forces on the body
are directly related to the change in
momentum of the fluid with time. The fluid momentum is equal to
the mass times the velocity of the fluid. Since the fluid is moving,
defining the mass gets a little tricky. If the mass of fluid were
brought to a halt, it would occupy some
volume
in space; and we could
define its density to be the mass divided
by the volume.
With a little math
we can show that the aerodynamic forces are
directly proportional
to the density of the fluid that flows by the
rocket. As a result of
this derivation, we also find that lift and drag depend on
the square of
the velocity.
Here is the derivation, beginning with Newton's second law:
F = d (m * V) / dt
where F is the force, m is the mass, t is time,
and V is the velocity. If we integrate this equation, we obtain:
F = constant * V * m / t
Since the fluid is moving, we must
determine the mass in terms of the mass flow rate.
The mass flow rate is the amount of mass passing a given point during
some time interval and its units are mass/time.
We can relate the mass flow rate to the density mathematically.
The mass flow rate mdot is equal to the density r
times the velocity times the area A through which the mass passes.
mdot = m / t = r * V * A
With knowledge of the mass flow rate, we can express the aerodynamic
force as equal to the mass flow rate times the velocity.
F = constant * V * r * V * A
A quick units check:
mass * length / time^2 = constant * length/time * mass/length^3 * length/time
* length^2
mass * length / time^2 = mass * length/time^2
Combining the velocity dependence and absorbing
the area into the constant, we find:
F = constant * r * V^2
The aerodynamic force equals a
constant times the density times the velocity squared. The
dynamic pressure
of a moving flow is equal to one half of the density times the velocity squared.Therefore, the aerodynamic force is directly proportional to the
dynamic pressure of the flow.
The velocity used in the lift and drag equations is the relative
velocity between an object and the flow. Since the aerodynamic
force depends on the square of the velocity, doubling the velocity
will quadruple the lift and drag.
You can investigate the effect of velocity and the other
factors on the flight of a rocket by using the
RocketModeler II Java Applet.
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