On the Rebound Which Follows the Percussion of Bodies




(1)
Mathematics in Naples, Naples, Italy

 


 Deceased



Some people think that the rebound which we observe is different from the movement of incidence of the same body. Some rebounds occur along the same straight line of incidence when this is perpendicular to the underlying plane which prevents its continuation. In such a case, the starting and the arriving points of the movement which they think must be distinct and opposite, are united in one and the same movement at the same point of incidence. These people also think that the first movement of incidence is completely annihilated by the resistance and firmness of the underlying plane and that afterwards an opposite movement of rebound is created anew. But the cause of this new movement is not easy to find. Indeed, as was shown above, the fluid air environment cannot carry out this movement nor the previous one. Thus, the obstacle constituted by the hard body necessarily must be the cause of this new movement of rebound. Actually this obstacle is either absolutely firm and immobile or otherwise it is shaken and quakes. If it is stable and absolutely deprived of movement, it cannot possibly produce a movement and an impetus by agitating and propelling another body so that the latter is able to rebound and resile. It then seems ridiculous to resort to the quaking of a marble wall since this quaking movement is very weak and thus unable to generate such velocity. Moreover, more elastic bodies carrying out more and longer quaking would repel more violently hard bodies in this incidence than an alpine rock does. This does not occur and is against evidence. Consequently, it must be admitted that the same impetus of the movement of incidence continues. Only because its passage and progression are prevented by an obstacle on its route, does it change its direction. This is a frequent habit in nature, as appears from this property of the pendulum which I discovered recently. Indeed, while a pendulum is oscillating, a nail is placed at mid-length of the rope, vertically below its attachment. The pendulum then is forced to oscillate along the circumference of a smaller circle, i.e. the previous impetus modifies its route and travels over a more curved route. Nevertheless, the distance travelled in the same time is equal very precisely to that through which it went over a larger circle. Moreover, the rebounds made obliquely at great angles on a plane or the ricochets on some curved surface obviously convince that those small ones result from the continuous action of the same movement in which the previous direction is in some way only returned by the curvature of the surface on which the body arrives or impinges. Thus, a body falling on some hard body is clearly impelled by the same motive virtue by which it will be moved afterwards by a normal necessity in rebounding through the shortest route.

But, although the resistance of a hard and immobile body does not annihilate completely the impetus of a body colliding with it, which is indicated by its rebound occurring after the incidence, one can at least wonder whether it weakens and in some way diminishes the impetus of the striking body. It is thus worthwhile to demonstrate at least what cannot happen.


Proposition LIX


The motive force of a striking body is neither weakened not diminished by the resistance of a firm and hard body.

Let a body A fall obliquely on a firm plane CD along the straight line AB (Table 7.​1, Fig. 46). Both bodies are hard, inelastic. A perpendicular AC is drawn to the underlying plane. This is divided into two equal parts BD and CB. DE is drawn perpendicular to the underlying plane and is equal to AC. B and E are joined by a straight line BE. Obviously, in the right-angled triangles ACB and EDB, since the sides of the right angles are equal, the angles ABC and EBD are equal. The body A falls obliquely on the plane BC. Thus, its movement through AB is the resultant of a double movement and impetus, one perpendicular AC and the other transverse CB. It is equal in power to these two movements. Therefore, the body A falls at B, provided with a double impetus. The transverse impulse is parallel to the plane CB and in no way compresses or strikes this plane. Thus, the underlying plane does not resist this impulse at all. It is as if the body was deprived of this impetus. Consequently, the perpendicular impetus alone remains able to compress and impel the underlying plane BC. The absolutely stable underlying plane must resist this impulse and can prevent its straight progression. Consequently, only the perpendicular movement undergoes resilience. It must be shown that this movement of rebound is neither diminished nor weakened. If possible, let the impetus of A, which is perpendicular and can be impeded, be diminished and weakened in the rebound by the resistance of the firm body CD. In the presence of a uniform movement, equal transverse distances CB and BD would be travelled in equal times since this movement does not undergo any alteration or diminution. But in the perpendicular movement of rebound it would travel a distance DF shorter than DE or AC because the perpendicular impetus is supposed to be diminished and weakened. Therefore, in the same time, it would travel the distance DF shorter than AC. This occurs in all the process of rebound. The course of this movement will be indicated by the oblique line BF, rather than BE, because the angle of rebound FBD will be smaller than the angle of incidence ABC. But this is untrue and against evidence. These angles indeed are always equal to each other. Consequently, the motive force of incidence of the body is neither weakened nor diminished by the resistance of a firm and hard body. Q.E.D.

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Oct 7, 2016 | Posted by in GENERAL | Comments Off on On the Rebound Which Follows the Percussion of Bodies

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