Page:Conservationofen00stew.djvu/35

Rh measurement, the kilogramme has now four times as much energy as it had in the last instance, because it can raise itself four times as high, and therefore do four times as much work, and thus we see that the energy is increased four times by doubling the velocity.

Had the initial velocity been three times that of the first instance, or 29.4 metres per second, it might in like manner be shown that the height attained would have been 44.1 metres, so that by tripling the velocity the energy is increased nine times.

28. We thus see that whether we measure the energy of a moving body by the thickness of the planks through which it can pierce its way, or by the height to which it can raise itself against gravity, the result arrived at is the same. We find the energy to he proportional to the square of the velocity, and we may formularize our conclusion as follows:—

Let $$v =$$ the initial velocity expressed in metres per second, then the energy in kilogrammetres $$= \frac{v^2}{19.6}$$. Of course, if the body shot upwards weighs two kilogrammes, then everything is doubled, if three kilogrammes, tripled, and so on; so that finally, if we denote by $$m$$ the mass of the body in kilogrammes, we shall have the energy in kilogrammetres $$= \frac{m v^2}{19.6}$$. To test the truth of this formula, we have only to apply it to the cases described in Arts 26 and 27.