268

THEORY AND PRACTICE

TRACT 37-

indeed quite equal, itsterminal velocity, suchas is peculiar to the par-ticular size anti weight ofthe ball, as assigned inprob. 10, which thereforeis the utmost limit of the■with.

velocity that the shot can descend

114. From this view of the ball’s motion, it is manifestthat the two branches of the path are both of them of thehyperbolic kind, having an asymptote, or tangent at an in-finite distance, in a particular situation, being oblique, asde, in the ascending branch, but perpendicular, as fg, inthe descending. Further, the velocity in the ascendingbranch is quicker, but the time in motion less, than in thedescending branch, as the motion in the latter can only ap-proach a certain limit, called the terminal velocity. Theascending branch too is less curved than the descending one.

115. Again, the slowest motion of the shot, in the para-bolic theory, is at the vertex ; but in the air, or resistancetheory, the point of slowest motion is a little beyond thevertex, in the descending branch, as at the point h; becausethe motion there, nearly horizontal, is much resisted by themedium, while it is but little assisted by the perpendiculardescent of gravity. For the same reason too, the point ofgreatest curvature will not be at the vertex, as in the para-bolic theory, but a little way beyond it, as at i. Further,the ordinate bc in the descending branch, is much less thanthat Ab in the ascending one, and both of them far less thanin the parabola, sometimes as much as in the proportion of10 or 20 to 1 ; the altitude bv is also much less.

llG. Lastly, the angle of elevation of the piece, to yieldthe greatest range or amplitude ac, with the same velocity,is not when it is equal to half the angle of the plane ac withthe vertical line ak, as in the parabola; but, on the contrary,is considerably less than half, and the more so both as theshot is smaller, and the velocity is greater; because, in the