An object that falls through a vacuum is subjected to only
one external force, the gravitational
force, expressed as the
weight
of the object. The weight
equation defines the weight W to be
equal to the mass of the object m times
the gravitational acceleration
g:
W = m * g
the value of g is 9.8 meters per square second
(32.2 feet per square second)
on the surface of the
Earth, and has different values on the surface of
the
Moon
and
Mars.
The gravitational acceleration g decreases with the square of
the distance from the center of the planet. But for many practical
problems, we can assume this factor to be a constant.
The mass of an object does not depend on the location, the weight does.
An object that moves because of the action of gravity alone is
said to be free falling.
If the object
falls through an atmosphere, there
is an additional drag force acting on
the object and the physics involved with
the motion of the object is more complex than in free fall.
For an object in free fall, we can easily predict the motion
of the object.
Assuming the mass of the object remains constant,
and the size and speed of the object is not so small or
so fast that we must consider relativistic effects,
the motion of the object is 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
For a free falling object, the net external force
is just the weight of the object:
F = W
Substituting into
the second law equation gives:
a = W / m = (m * g) / m = g
The acceleration of the object
equals the gravitational acceleration. The mass, size, and shape of
the object are not a factor in describing the motion of the object.
So all objects, regardless of size or shape or weight,
free fall with the same acceleration. On the figure, we show an orbiting
Space Shuttle and a space walking astronaut. The astronaut and the Shuttle
have very different weight, size and shape.
But objects in orbit are in a free fall and the
only force acting on the objects is the gravitational attraction of
the Earth. So both the astronaut and the Shuttle are accelerated
towards the Earth with the same acceleration.
Because the objects orbit at some altitude above the Earth's
surface, the acceleration is slightly less than the surface value.
At a 200 mile orbit the acceleration is about 90% of the surface value.
Since both Space Shuttle and astronaut are falling with the same acceleration,
the astronaut appears to be "weightless" and "floats" relative to the Shuttle.
If you know the local value of the gravitational acceleration, you
can use the equations for
translation
of an object to obtain the instantaneous velocity and location as
a function of time. The mass must remain
constant for a constant acceleration to occur.
If one launches an object from the surface of a planet, and there
is only gravity acting on the object (no thrust and no drag), the
resulting
trajectory
is described by the ballisitc flight equations.
The remarkable observation that all free falling objects fall
with the same acceleration was first proposed by
Galileo, nearly 400 years ago.
Galileo conducted experiments using a ball on an inclined plane
to determine the relationship between the time and distance traveled.
He found that the distance depended on the
square of the time
and that the velocity increased as the ball moved down the incline. The
relationship was the same regardless of the mass of the ball used in
the experiment. The story that Galileo demonstrated his findings by
dropping two cannon balls off the Leaning Tower of Pisa is just a
legend.
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