Lesson 2 The Law of Falling Bodies

Galileo Galilei proposed that all bodies fall with the same constant acceleration because the effect of gravity is the same on all objects. However, on Earth, air resistence affects the rate at which objects fall. This is why a feather falls to the ground at a slower rate than a hammer on Earth, but in a vacuum, a volume of space that is empty of matter, both the hammer and feather would hit the ground at the same time. Galileo's theory was tested on the Apollo 15 mission, part of the American Apollo space program by David R. Scott. Scott held both a hammer and feather at the same distance from the ground and released them at the same time. The moon served as a perfect vacuum and the hammer and feather in fact landed on the surface of the moon at the same time. Galileo was correct!

Galileo not only proposed that bodies fall with the same constant acceleration but that their distance fallen is proportional to the odd numbers. This theory was known as the Law of Odd Numbers. This theory is displayed in the image below:

After each time period the object falls in an odd numbered interval. Additionally, the total distance fallen after each interval is equal to the perfect square. In other words, "distance fallen is proportional to the square of time." From this we can form the equation:

S(t) = ct^(2)

Where (c) represents a constant of about sixteen feet and (t) represents time, this equation can be used to find the distance traveled in a particular amount of time. For example, if a woman is on an amusement park ride and is free falling, what is her distance fallen after two seconds?:

S(2) = c(2)^(2)

S(2) = 4c

S(2) = 4(16)

S(2) = 64 ft.

The woman falls sixty-four feet after two seconds, but what is her speed after two seconds? The woman's speed can be calculated by using the equation:

S = d/t

Where (d) represents distance and (t) represents time.

S= 64ft./2sec.

S=32ft./sec. 

We have just found the woman's instantaneous speed, but what if we want to find her average speed as she is free falling on the ride from time (t) to time (h) seconds later?

Average Speed = Change is Distance

                            Change in Time

S(t) = ct^2

S(t+h) = c(t+h)^2

Average Speed = S(t+h) - S(t)

                             h

                           = c(t+h)^2-ct^2

                             h

(t+h)^2 = t^2+2th+h^2

c(t+h)^2 = ct^2+2cth+ch^2

                                      = ct^2+2cth+ch^2-ct^2

                                         h

                     = 2cth+ch^2

                        h

Average Speed = 2ct+ch

Instantaneous Speed = 2ct, therefore V(t) = 2ct (V represents Velocity)

Distance = S(t) = 16t^2 and V(t) = 32t

Thus speed (velocity) is the derivative of distance.

Now we must find the acceleration of the woman, which according to Galileo should be constant.

V(c) = 2ct

V(t+h) = 2c(t+h)

            = 2ct+2ch

Average Acceleration = 2ct+2ch-2ct

                                       h

                     = 2c

                                 A(t) = 2c (Always the same)

This mathematically proves that Galileo's theory was correct. Since A(t) = 2c and the force of gravity is always constant:

The force of Gravity (g) is equal to 2c

g = 2c

c = 1/2g

Distance: S(t) = 1/2gt^2

Speed (Velocity): V(t) = gt

Acceleration: A(t) = g 

As displayed above, derivatives are a vital component of physics. Ultimately accelerated motion is uniform by the odd numbered law proposed by Galileo.