It will be noticed from the above that the velocity is proportional to the number of seconds, but that the distance traveled is proportional to the _square_ of the number of seconds, and also proportional to the square of the velocity. Momentum is mass multiplied by velocity; energy is measured by the distance through which a body will move against a given resistance. Should you prop up one wheel of a carriage and revolve the wheel, then with the pressure of the finger or the thumb on the hub as a brake, stop it, it will be found that (omitting the effect of atmospheric resistance), the wheel will make four times as many revolutions before stopping with a doubled velocity; nine times as many with a trebled velocity. Falling bodies afford the most perfect illustration of the principle of Momentum and Energy, and are so commonly used to illustrate those principles that many students get the idea that the application of those principles is confined to falling bodies, and do not realize that they extend generally through the field of mechanics. A falling body is, of course, acted upon by gravity with uniform force equal to the weight of the falling body, and that force continues to follow the falling body and to be applied uniformly and equally, however slowly, or rapidly the body may be falling. And, omitting atmospheric resistance, the body is absolutely free to move except for its natural tendency to remain at rest, or at uniform velocity. It is well known that a body falls (almost exactly) sixteen feet in one second, and at the end of one second has a velocity of thirty-two. During the second second it falls through a distance of forty-eight feet, and during the third second a distance of eighty feet. In two seconds it falls sixty-four feet, and in three seconds one hundred twenty-eight feet, and so on. Thus, it will be observed that the _velocity_ is proportional to the time during which it has fallen, but that the distance fallen in any number of seconds is proportional to the _square_ of the time. This, indeed, is a property of numbers, and results from mathematical law. If the reader will form a series of numbers, setting down any number for the first term of the series, adding to it its double for the second term, and adding to the second term double the first term for the third, and adding double the first term to the third term for the fourth, and so on--in other words, form any increasing arithmetical series with double the first term for the common difference, he will discover that the _sum of all the terms is equal to the first term multiplied by the square of the number of terms_. Thus: 1st Term 2nd Term 3rd Term 4th Term 5th Term 5 15 25 35 45