TRACT 34.

IN GUNNERY.

333

wood is, and therefore to whatever depth the ball pene-trates, and the same as if the. wood were perfectly hard, orthe ball made no penetration at all. And this velocity of thepoint of impact also agrees with that which was found inArt. 21. So that the velocity communicated to the point ofimpact is the same, whether the impulse is made in aninstant, or in some small portion of time. And hence, inthe usual case of a penetration, because the block will havemoved some small distance before it has attained its greatestvelocity, it might at first view seem as if it would swing orrise higher than when that velocity is communicated inan instant, or when the pendulum is yet in its vertical posi-tion, and so might describe a longer chord, and show agreater velocity of the ball than it ought. But, on theother hand, it must be considered, that in the small part ofits swing, which the pendulum has made before the pene-tration is completed, or has attained its greatest velocity,just as much velocity will be lost by the opposing gravityor weight of the pendulum, as if it had set out from thevertical position with the said greatest velocity ; and there-fore the real velocity at that height will be the same in bothcases. Hence then it may safely be concluded, that thecircumstance of the ball’s penetration causes no alteration inthe velocity of it, as computed by our formula. And as it•was before found that no sensible error is incurred by thetwo first circumstances, namely, the friction on the axis,and the resistance of the air to the back of the pendulum,we may be well assured that our formula brings out the truevelocity with which the ball strikes the pendulum, withoutany sensible error.

29. Since

bi iv

. denotes the greatest velocity which

figO + Oil *- J

the point c of the pendulum acquires by the stroke, divid-ing by i, we shall have ■ ;^ for the angular velocityof the pendulum, or that of a radius 1. From which itappears, that the vibration will be very small when i or thedistance A d is small, and also when i is very great. And if