390 FRAGMENTS OF SCIENCE 



of work done in all these cases, as far as the raising of 

 the weight is concerned, would be absolutely the same. 

 The work done at one and the same 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 A, then the product of m into A, or m h, 

 expresses, or is proportional 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 82 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 start- 

 ing, 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 starting, 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 be- 

 fore 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 pro- 

 portional to or represented by m v\ In the case consid- 

 ered, I have supposed the weight to be cast upward, 



