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The Acceleration Of A Freely Falling Body

1729 words - 7 pages

The Acceleration of a Freely Falling Body

To study the motion of a freely falling body, an object is allowed to
fall and its position after successive equal time intervals is
recorded on wax-coated paper by means of electric sparks. From these
data, graphs of distance vs. time and velocity vs. time are plotted.
The acceleration due to gravity is found by determining the slope of
the velocity vs. time graph.

Theory

In one dimension, an object's average velocity over an interval is the
quotient of the distance it travels and the time required to travel
that distance:

(1)

where and . The instantaneous velocity at a point is defined as the
limit of this ratio as the time interval is made vanishingly small:

(2)

Hence, the velocity is given by the slope of the tangent to the
distance vs. time curve. If the velocity were constant the slope would
be constant, and the curve would be a straight line. This is evidently
not the case for a freely falling body, since it is at rest initially
but has nonzero velocities at later times.

When the velocity of a body varies, the motion is said to be
accelerated. The average acceleration over an interval is the quotient
of the change of the instantaneous velocity and the time required for
that change:

where . The instantaneous acceleration is defined analogously to the
instantaneous velocity:

(3)

If a body moves in a straight line and makes equal changes of velocity
in equal intervals of time, the body is said to exhibit uniformly
accelerated motion. This type of motion is produced when the net force
upon a body is constant. An example of this is the motion of a body
falling freely in a vacuum. The acceleration of the body is called the
acceleration due to gravity, g, and has the approximate value of 9.81
m/s2 (= 981 cm/s2 = 32.2 ft/s2) near the surface of the earth.

For uniformly accelerated motion (a = constant), the instantaneous
acceleration is given by (3), which can be rearranged to give

When this equation is integrated from time to to t where the
respective velocities are vo and v, the result is

(4)

or

The graph of velocity vs. time is thus a straight line, the slope of
which is the acceleration, a. In the experiment the value of g will be
determined using this fact.

When (4) is substituted into (2) and the resulting equation is
rearranged, the result becomes

The value of this expression when integrated from to to t, where the
respective displacements are so and s, is

This equation shows that the distance vs. time graph is parabolic.

An important fact which will be used in graphing velocity vs. time is
that for motion with constant acceleration, the average velocity
between two displacements equals the instantaneous velocity at the
midpoint in time of the interval. That...

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