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Introduction

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.

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Galileo Galilei versus 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.

 

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Galileo Galilei Law of Free Fall

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/s2, 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

g=9.8 m/s^2 always vertical down

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.

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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.

 

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by Luis F. Sez, Ph. D.    Comments and Suggestions: LSaez@dallaswinwin.com