370 FBAGMENTS OF SCIENCE. 



place, and neglecting the small change of gravity with 

 the height, depends solely upon two things; on the 

 quantity of matter lifted, and on the height to which 

 it is lifted. If we call the quantity or mass of matter 

 m, and the height through which it is lifted li, then the 

 product of m into Ji, or m h, expresses, or is propor- 

 tional to, the amount of work done. 



Supposing, instead of imparting a velocity of 32 

 feet a 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 quad- 

 ruple the speed at starting, we attain sixteen times the 

 height. Thus, with a four-fold velocity of 128 feet a 

 second at starting, the weight would attain an elevation 

 of 256 feet. With a seven-fold velocity at starting, the 

 weight would rise to 49 times the height, or to an ele- 

 vation 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 proportional to or repre- 

 sented by in v 2 . In the case considered, I have sup- 

 posed 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 ex- 



