Galileo's law of odd numbers

In classical mechanics and kinematics, Galileo's law of odd numbers states that the distance covered by a falling object in successive equal time intervals is linearly proportional to the odd numbers. That is, if a body falling from rest covers one unit of distance in the first arbitrary time interval, it covers 3, 5, 7, etc. units of distance in subsequent time intervals of the same length. This mathematical model is accurate if the body is not subject to any forces besides uniform gravity (e.g. it is falling in a vacuum in a uniform gravitational field). This law was established by Galileo Galilei who was the first to make quantitative studies of free fall.

Explanation
Derivation of Galileo's law of odd numbers

Using a speed-time graph

The graph in the figure is a plot of speed versus time. Distance covered is the area under the line. Each time interval is coloured differently. The distance covered in the second and subsequent intervals is the area of its trapezium, which can be subdivided into triangles as shown. As each triangle has the same base and height, they have the same area as the triangle in the first interval. It can be observed that every interval has two more triangles than the previous one. Since the first interval has one triangle, this leads to the odd numbers.[1]

Using the sum of first n odd numbers

From the equation for uniform linear acceleration, the distance covered s = ut + 1/2at2, for initial speed u = 0, constant a (acceleration due to gravity without air resistance), and time elapsed t, it follows that s ∝ t2, thus the distance from the starting point are consecutive squares for integer values of time elapsed. The middle figure in the diagram is a visual proof that the sum of the first n odd numbers is n2.[2] In equations:

1= 1= 12
1 + 3= 4= 22
1 + 3 + 5= 9= 32
1 + 3 + 5 + 7= 16= 42
1 + 3 + 5 + 7 + 9= 25= 52

Schroedinger states in his book Mind and Matter that the pattern is believed to continue to infinity, though it has never been proven to do so.

See also


  1. Galelio Trick
  1. S Ducheyne, Galileo and Huygens on free fall: Mathematical and methodological differences
  2. RP Olenick et al., The Mechanical Universe: Introduction to Mechanics and Heat

Vsauce explains The Odd Number Rule


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