236

THEORY AND PRACTICE

tract 37 -

velocity, between the greatest velocity and nothing, fromwhich to compute the other general resistances, by consider-ing them as the squares of the velocities. It is proper to as-sume a near medium velocity and its resistance, because, ifwe assume or commence with the greatest, or the velocity ofprojection, and compute from it downwards, the resistanceswill be everywhere too great, and the altitude ascended muchless than just; and, on the other hand, if we assume or com-mence with a small resistance, and compute from it all theothers upwards, they will be much too little, and the com-puted altitude far too great. But, commencing with a me-dium degree, as for instance that which has a resistanceabout the half of the first or greatest resistance, or rather alittle more, and computing from that, then all those com-puted resistances above that, will be rather too little, but allthose below it too great; by which it will happen, that thedelect of the one side will be compensated by the excess onthe other ; and the final conclusion be near the truth.

42. Thus then, if we wish to determine, in this way, thealtitude ascended by the hall employed in the former tableof resistances, when projected with 2000 feet velocity; weperceive by the table, that to the velocity 2000 correspondsthe resistance 102lh ; the half of this is 51, to which resist-ance corresponds the velocity 1400 in the table, and the nextgreater velocity 1500, with its resistance 59, will be proper-est to be employed here. Hence then, for any other velocityv, in general, it will be, according to the law of the squares

of the velocities, as 1500* : i* 1 : : 59 : = '000026|i; 1 =

av \ putting a = -000026!, which will denote the air’s resist-ance for any velocity v, very nearly, counting from 2000down to 0.

43. Now let x denote tile altitude ascended when the ve-locity is v, and W the weight of the ball: then, as above, av 1is the resistance Irorn the air, hence av z -j- w is the whole rc-

• • r 1 + W r i ..

sisting force, and —— = J tne retarding force ;