THEORY AND PRACTICE
TRACT 31,
243
.those velocities, or the heights from which the halls must de-scend in vacuo to acquire them. Also the 5th or last columnshows the times of descending through these heights freely,or without resistance.
But it is manifest that the balls can never attain exactly tothese velocities in any finite time or descent, being only thelimits to which they continually approach, without ever reallyreaching, though they arrive very nearly at them in a shortspace of time; as will appear hereafter.
70. The greatest velocity being 17 Sv'd feet, when d isexpressed in inches, it will be 118v / l2ri or Q\G'QVd, whered is expressed in feet. Now, the spaces through which abody must freely descend by gravity, to acquire the samevelocity, will be thus found : as S2 : V 16 : : OlG'fi V d : \Zh,or 8:1:: 016-6 v / r?: ^/h, or 77" 1 Vd — V/i, hence h ~5944 d. But the density of the ball is to the density of theair, as 7400 to 1£ nearly, or as 6G600 to 11, or as 6054 to I.And 6054 to 1 is nearly the same ratio as 5944<i or h to d.Therefore the greatest velocity with which a ball can descendin the air, is the same which it may acquire by falling with-out resistance, and in its fall describing a space that is ipproportion to the ball’s diameter, as the density of the ballto the density of the air, very nearly.
The exact spaces, h or 5944r/, answering to the velocitiesv in the 3rd column, are set in the 4th column of the tableabove. Also the times of freely falling in vacuo throughthose spaces, or of acquiring the. velocities in the 3rd column,are placed in tiie 5th or last column, being found by dividingthe greatest velocity v by 32 feet, the velocity acquired inthe first second of time, viz, — = / seconds, the time.
71. To obtain general expressions for the space descended,and the time of the descent, in terms of the velocity v: putjc = any space descended, t — its time, and v the velocityacquired, the weight of the ball ti; — i|1b. Now, by thetheorem near the end of prob. 2, which is the proper rule forthis case, the velocity being small, ■0000176V' = cv 2 is the re-sistance due to the velocity v ; tlicrcf. w — cv~ is the impelling