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height; in like manner, a threefold velocity will give a
ninefold elevation, a fourfold velocity will give a sixteen-
fold elevation, and so on. The height attained, then, or the
work done, is not proportional to the velocity, but to the
square of the velocity. As before, the work is also pro-
portional to the weight elevated. Hence the work which
any moving masses whatever are competent to perform, by
the motion which they at any moment possess, is jointly
proportional to the weight and the square of the velocity.
Here, then, we have a second measure of work, in which we
simply translate the idea of height into its equivalent idea
of motion.

In mechanics, the product of the mass of a moving body
into the square of its velocity, expresses what is called the
vis viva, or living force. It is also sometimes called the
“mechanical effect.” If, for example, we point a cannon
upward, and start a ball with twice the velocity imparted
by a second cannon, the ball will rise to four times the
height. The Speedier ball, if directed against a target, will
also do four times the execution. Hence the importance
of imparting a high velocity to projectiles in war. Having
thus cleared our way to a perfectly clear conception of the
vis viva of moving masses, we are prepared for the an-
nouncement that the heat generated by the collision of a
falling body against the earth is proportional to the vis
viva annihilated. In point of fact it is not an annihilation
at all, but a transference of vis viva from the mass, to its
ultimate particles. This, as we now learn, is proportional
to the square of the velocity. In the case, therefore, of two
cannon-balls of equal weight, if one strike a target with
twice the velocity of the other, it will generate four times
the heat; if with three times the velocity, it will generate
nine times the heat, and so on.

Mr. Joule has shown that in falling from a height of 772
feet, a body will generate an amount of heat sufficient to

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