The greater the velocity, the more energy is required to impart a given acceleration. To increase the velocity from ten feet per second to twenty feet per second, the applied force must continue through one second of time, and more energy is required to follow a rapidly moving body, and continue to apply to it a given force for one second than would be required to follow and maintain the application of the same force to a body moving more slowly—the distance traveled is greater in one case than in the other.
It must be plain that if the moving body have a velocity at the end of the first second of twenty feet per second, it will, at the end of the second second, with the same pressure (force) continued against the same resistance, have a velocity of forty feet per second, and at the end of three seconds have a velocity of sixty feet, and at the end of four seconds a velocity of eighty feet, and so on.
Now, at the beginning of the second second it had a velocity of twenty feet, and at the end of that second a velocity of forty feet. It therefore, traveled through that second with an average velocity of thirty feet and, of course, during the second second traveled exactly thirty feet. It traveled ten feet the first second, and if it traveled thirty feet the second, then in the two seconds it traveled forty feet—four times as far as it traveled the first second. At the beginning of the third second it had a velocity of forty feet, and at the end of the third second a velocity of sixty feet. The average velocity then for the third second would be one-half the sum of forty feet and plus sixty feet—that is to say, it would be fifty feet, and that would be the distance traveled during the third second. The first second it traveled ten feet, the second second thirty feet, and the third second fifty feet, making a total in three seconds of ninety feet—that is to say, in three seconds it traveled nine times as far as in one second.
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 |