(1)
Mathematics in Naples, Naples, Italy
Deceased
From what was said above it is clear that two bodies colliding in opposite movements sometimes lose their impetus completely and are reduced to immobility, sometimes their velocities remain absolutely unaltered and of the same vigour. But it seems impossible that the impulses, after being weakened or extinguished in the collision of the bodies, revivify afterwards since one and the same cause, namely the repulsion of an opposite movement, cannot generate immobility and movement in the same subject. But it is obvious that this occurs in elastic and resilient bodies such as are machines and other bodies of the same nature. Indeed, a playing ball is at first slowed down by an elastic net and then very impetuously propelled. A ball inflated with air or a water-skin, like other machines which are constricted and dilated, is also at first slowed down and afterwards very impetuously impelled. In all of these examples one of the two bodies is immobile. It is either the ball or the racket. After the immobility a well-determined velocity is transmitted in a straight course when an immobile ball is struck, or moved by the impetus of a recoil, when the ball strikes an immobile net. Then a movement of recoil is generated. Actually, if the ball on one hand and the net on the other hand collide in opposite movements, after slowing down or extinction of the movement, the percussion which occurs is stronger by as much as the two velocities, that of the ball and that of the net, together are greater than the impetus of one of them alone. The same clearly occurs in similar machines. It is thus worth-while to search for and to explain the cause of this effect.
Proposition LXXII
An elastic rod is fixed at right angles on an underlying plane. If impelled transversely by some body, the rod while bending extinguishes the percussive impetus but generates it anew while resiling.
Let an elastic rod EB be fixed to the ground RS at E (Table 16.1, Fig. 57). Instinctively of course it remains spontaneously immobile perpendicular to the horizon since it is balanced by equal forces, i.e. by its fibres parallel to its base FE. Every of them plays the role of a lever and they prop up each other so that they resist lateral bending to any side, towards G or towards I. These fibres, however, are united and tied, not tightly, but by some slippery connection so that they can yield to any impulse. If displaced from its perpendicular position EB towards G, the rod EB resists this displacement the more, the more it moves away from C. This resistance is not purely passive and inert such as that found in transported bodies which, wherever they are set down, remain immobile and do not try to return to their previous position. The opposite occurs in the rod EB which always attempts with more and more energy at coming back to C, the more so the more it is displaced towards G. This effort is carried out with such energy that the eagerness to retrocede is not satisfied with returning to the position perpendicular at C, and the rod moves further, going beyond the perpendicular, over the arc CI. The projected body A is supposed to move from A towards C at a velocity D and to strike the top of the rod B. Consequently, a first percussion is carried out at C and the rod is pushed towards G. After the first collision, while the rod leaves the perpendicular position, the receding action by which it resists the impulsive velocity of A begins. The impetus D is the more diminished and the progress of its diminution is the more pronounced, the more the rod EB together with A moves away from the perpendicular. Therefore, the velocity D is extinguished at G. Thus, the body A brought to the position G together with B appears to have lost all its impetus D. But the rod EB at G is not equally deprived of impetus and motive virtue. It tries to return to its perpendicular position with as much energy as is the velocity D, i.e. as is exactly the force by which it was expelled from its perpendicular position. This energy is such that EG can recede over a period of time equal to that during which it was expelled. The impulses are continued in repelling the annexed body A and in restoring in it or rather in generating anew a degree of velocity equal to D, which it had consumed when arriving. Consequently, in the perpendicular position C the body A is impregnated with the same degree of velocity D with which it had arrived there. Thus, it necessarily resiles and recoils in an opposite movement from C towards A at the same velocity at which it had arrived there.
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