An object that is falling through the
atmosphere
is subjected to two external
forces.
The first force is the gravitational force, expressed as the
weight
of the object, and the second force is the aerodynamic
drag
of the object. The
weight equation
defines the weight W to be
equal to the mass m
of the object times the gravitational acceleration
g:
W = m * g
the value of g is 9.8 meters per square second on the
surface of the earth. The gravitational acceleration decreases with
the square of the distance from the center of the earth.
But for most practical problems in the atmosphere, we can assume this
factor is constant. If the object were falling in a vacuum,
this would be the only
force
acting on the object. But in the atmosphere, the motion of a falling
object is opposed by the aerodynamic
drag.
The
drag equation
tells us that drag D is equal to a
drag coefficient Cd
times one half the air density r
times the
velocity V
squared times a reference
area A
on which the drag coefficient is based:
On the figure at the top, the density is expressed by the Greek symbol
"rho". The symbol looks like a script "p". This is the standard symbol used by
aeronautical engineers. We are using "r" in the text for ease of translation
by interpretive software.
D = Cd * .5 * r * V^2 * A
The motion of any moving object can be described by Newton's
second law
of motion, force F equals mass m
times acceleration a:
F = m * a
We can do a little algebra and solve for the
acceleration of the object in terms of the net external
force and the mass of the object:
a = F / m
Weight and drag are forces which are
vector quantities.
The net external force is then equal to the
difference
of the weight and the drag forces:
F = W - D
The acceleration of the object then becomes:
a = (W - D) / m
The drag force depends on the square of the velocity.
So as the body accelerates its velocity and the drag increase.
It quickly reaches a point
where the drag is exactly equal to the weight.
When drag is equal to weight, there is no net external force
on the object, and the acceleration becomes zero.
The object then falls at a constant velocity as described by
Newton's
first law
of motion. The constant velocity is called the
terminal velocity.
Activities:
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Falling Objects:
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