282 - FRAGMENTS OF SCIENCE. 
second we impart at starting twice this velocity. To what 
height will the weight rise? You might be disposed to 
answer, " To twice the height?" but this would be quite 
incorrect. Instead of twice 16, or 32 feet, it would reach 
a height of four times 16, or 64 feet. So also, if we treble 
the starting velocity, the weight would reach nine times 
the height; if we quadruple the speed at starting, we attain 
sixteen times the height. Thus, with a fourfold velocity 
of 128 feet a second at starting, the weight would attain an 
elevation of 256 feet. With a sevenfold velocity at start- 
ing, the weight would rise to 49 times the height, or to an 
elevation of 784 feet. 
Now the work done or, as it is sometimes called, the 
mechanical effect other things being constant, is, as before 
explained, proportional to the height, and as a double 
velocity gives four times the height, a treble velocity nine 
times the height, and so on, it is perfectly plain that the 
mechanical effect increases as the square of the velocity. 
If the mass of the body be represented by the letter m, and 
its velocity by v, the mechanical effect would be propor- 
tional to or represented by m v z . In the case considered, 
I have supposed the weight to be cast upward, being 
opposed in its flight by the resistance of gravity; but the 
same holds true if the projectile be sent into water, mud, 
earth, timber, or other resisting material. If, for example, 
we double the velocity of a cannon ball we quadruple its 
mechanical effect. Hence the importance of augmenting 
the velocity of a projectile, and hence the philosophy of 
Sir William Armstrong in using a large charge of powder 
in his recent striking experiments. 
The measure then of mechanical effect is the mass of the 
body multiplied by the square of its velocity. 
Now in firing a ball against a target the projectile, after 
collision, is often found hot. Mr. Fairbairn informs me 
that in the experiments at Shoeburyness it is a common 
thing to see a flash, even in broad daylight, when the ball 
strikes the target. And if our lead weight be examined 
after it has fallen from a height it is also found heated. 
Now here experiment and reasoning lead us to the remark- 
able law that, like the mechanical effect, the amount of 
he-it generated is proportional to the product of the mass 
into the square of the velocity. Double your mass, other 
things being equal, and you double your amount of heutj 
