

A practical application of motion with constant acceleration
is the motion called free fall. The statement free fall means object falling in
absence of air resistance.
WRONG THEORY: The study of
falling objects was first formalized by Aristotle who based his conclusions on
the theory of the four elements, air, water, fire, and Earth. Under this theory,
different material will act accordingly with their nature; a rock will fall
because is made out of Earth elements and the natural tendency of such material
is to be with materials of the same nature. In the same form, smoke will have to
raise up toward the natural place for air elements. Materials with combinations
of elements such as a fader will fall to the ground at a different speed because
of its tendency to belong to the two elements, air and Earth. Among the
conclusions of Aristotle's theory is that object with different weights will
fall with different rates with heavier objects falling faster than lighter
objects. Galileo Galilei will be the first scientist to prove this theory wrong.
Aristotle lived between 384 and 322 B. C. while Galilei lived between 1564 and
1642. Therefore, between their lives there are more than 17 centuries!
Galileo introduced the modern methodology of the scientific
method which bases the analysis of scientific hypothesis in control
experimentation designed to prove or disprove the validity of the proposed
hypothesis. In this form, theories based on the wrong hypothesis will not
survive for long period of time as it did the theory of falling objects proposed
by Aristotle.


In addition to performing innumerable experiments involving
falling objects, Galileo Galilei presented the following argument to prove
contradictory the conclusions of Aristotle hypothesis that objects with
different weights falls at different rates. The discussions below will neglect
the effect of air resistance. Even when air resistance is present for most
falls, eliminating the effect of air on the motion of objects allows for a first
approximation of this general motion.

In the applet on the left, object A and B are of the
same material; therefore, because of the difference in side, object A is
heavier than object B.
Based on Aristotle theory, which of the two objects shown in
Part I of the applet reaches the ground
first?
N 
a) 
A 


b) 
B 

c) 
Both at the same time 

d) 
Not enough Information 

e) 
None of the above. 

Part II of the applet shows
what would happen if Aristotle theory is correct; i. e., object A
reaches the ground first because is heavier than object B.
Now, for Part III
of the applet, objects A and C are identical to each other,
accordingly with Aristotle theory of falling object, which one
should reach the ground first, A or C?

a) 
A 


b) 
C 
N 
c) 
Both at the same time 

d) 
Not enough Information 

e) 
None of the above. 

As shown in Part IV
of the applet, both object should reach the ground at the same
time because the two objects have the same weight. Aristotle
theory indicates that object reach the ground accordingly with
their weights but, in this case, the objects have the same
weight and, consequently, they reach the ground at the same time
even when Aristotle theory is applied. 
Again, Part V of
the applet shows a situation where object B has been tied with a
rope to object A. The combination of objects A and B and object
C are drop simultaneously from the same height. Using Aristotle
theory of falling objects, which one reaches the ground first, A
and B or C?

a) 
A and B, because together they are heavier than C. 


b) 
C because B drags behind object A slowing it down 

c) 
Based on Aristotle theory, both the combination of
objects A and B and object C reach the ground at the same time. 
N 
d) 
Based on Aristotle theory, options a) and
b) are correct implying a contradiction in the theory. 

e) 
None of the above. 

Part VI and
Part VII of the applet show two different
results that can be concluded from Aristotle theory of free fall.
Part VI shows the heavier combination of A
and B reaching the ground first in accordance with Aristotle theory.
However, Part VII shows object C reaching
the ground first because the combination A and B is slow down by object
B which, accordingly with Aristotle theory, falls slower than either A
or C. In fact, B acts like a parachute even when there is not air
resistance on this analysis (air resistance is neglected). Since the two
conclusions are derived from the same theory, the theory produce a
contradiction. This contradiction is associated with the theory assuming
that object fall to the ground at rates that depends on the weight of
the objects. 


Again, free fall is the study of how objects fall in the
absence of air resistance. After Galileo Galilei performed innumerable
experiments involving the falling of objects, he reached the following
experimental conclusion:
In the absence of air resistance, all objects fall with
the same constant acceleration, g
= 9.8 m/s^{2}, this acceleration is called the acceleration of
gravity. This acceleration is always pointing downward independent of the
direction of motion of the object (upward or downward).

Notice that Galileo concluded that the acceleration
is common to all falling objects not so the velocity. In fact objects
may have different velocities depending of the initial conditions of the
motion. Nevertheless, independent of those initial conditions and the
motion being upward or downward all objects have the same constant
acceleration. Locally
Vectors pointing down are considered negative;
therefore, the acceleration of an object in free fall is:
The index y will be used to identified the
vertical axis. 
At Galileo's era, the value of the acceleration of gravity
could only be measured from experiments and could not be derived from the
existing theories; therefore, this result is called a law
of nature (it can only be confirmed through experimental results). Later,
after
Newton's gravitational law, the value of the
acceleration of gravity can be calculated from other constants such as the mass
and radius of the Earth. At that time, the acceleration of gravity is not longer
a law of nature but the result of a more general law of nature as is Newton's
gravitational law.

Equations of Motion for Free Fall 
Since this motion is with constant acceleration, the
equations for free fall are the same as the equation for uniform accelerated
motion. However, free fall is a motion along the vertical rather than the
horizontal. Therefore, the displacement will be identified with the variable
y rather than x. The following are the equations for free fall after
the corresponding changes have been made:
Acceleration 

The acceleration of gravity is always pointing down.
The downward direction will be associated with the negative sign in most
cases for these notes. 
Velocity (average) 

(F_F 1) The index y
is used to emphasize the vertical nature of the motion. 
Velocity (average) 

(F_F 2) 
Velocity (instantaneous) 

(F_F 3) 
Displacement 

(F_F 4)
Again, this
equation is for the displacement and not the distance traveled. Also,
this equation assumes that the origin of the y axis is at the
level where the motion starts at t = 0. 
Time Independent Equation 

(F_F 5) 
The previous group of equations is used similarly than the
equations for uniform accelerated motion. In this section of the notes,
additional applications of free fall will be used.
