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THEORY AND PRACTICE

TRACT 37.

urging force, being always the same, and producing an equalincrease of velocity in equal times, excepting for the diminu-tion of motion by the air’s resistance. It is also evident thatthis resistance, beginning from nothing, continually increases,in some ratio, with the increasing velocity of the ball. Now,as the urging force is constantly the same, and the resistingforce always increasing, it must happen that the latter willat length become equal to the former : when this obtains,there can afterwards be no further acceleration of the mo-tion, the impelling force and the resistance being equal, andthe ball must ever after descend with a uniform motion. Itfollows therefore that, to answer the first enquiry, we haveonly to determine when or what velocity of the ball willcause a resistance just equal to its own weight.

67. Now, by inspecting the table of resistances precedingprob. 1, or in prob. 2, the weight of the ball being 1-j-lb, weperceive that the resistance increases in the last column, till0'7 09 opposite to 200 velocity, and 1-612 answering to 300velocity, between which two the proposed resistance 1-125,and the correspondent velocity, fall. But, in two velocitiesnot greatly different, the resistances are very nearly propor-tional to the squares of the velocities. Therefore, havinggiven the velocity 200 answering to the resistance 0-709, tofind the velocity answering to the resistance 1*125, we mustsay, as 0-709 : 1-125 : : 200* : v z = 63470, therefore v =\/63470 = 252, is the greatest velocity this ball can acquire;after which it will descend with that velocity uniformly, or atleast with a velocity nearly approaching to 252.

The same greatest or uniform velocity will also be directlyfound from the rule -0000176©* x= r, near the end of. pro-blem 2, where r is the resistance to the velocity v, by making

1-125 = T; for then ©* = -oooom = 6 3920, the root of whichis 253, the same value as before nearly.

68. But now, for any other weight of ball; since theweights of the balls increase as the cubes of their diameters,and their resistances, being as the surfaces, increase only as