370 FRAGMENTS OF SCIENCE. 



lutely 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 h, then the 

 product of m into h, or tn A, 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 dis- 

 posed to answer, ' To twice the height ; ' but this would 

 be quite incorrect. Instead of twice 1 6, 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 four- fold velocity of 128 feet a second at start- 

 ing, 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 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 proportional to or represented by m v 2 . 

 In the case considered, I have supposed the weight to be 

 cast upward, being opposed in its flight by the resist- 

 ance of gravity ; but the same holds true if the pro- 

 jectile be sent into water, mud, earth, timber, or other 



