PERCUSSION
, in Physics, the impression a body makes in falling or striking upon another; or the shock or collision of two bodies, which meeting alter each other's motion.
Percussion is either Direct or Oblique. It is also either of Elastic or Nonelastic bodies, which have each their different laws. It is true, we know of no bodies in nature that are either perfectly elastic or the contrary; but all partaking that property in different degrees; even the hardest and the softest being not entirely divested of it. But, for the sake of perspicuity, it is usual, and proper, to treat of these two separately and apart.
Direct Percussion is that in which the impulse is made in the direction of a line perpendicular at the place of impact, and which also passes through the common centre of gravity of the two striking bodies. As is the case in two spheres, when the line of the direction of the stroke passes through the centres of both spheres; for then the same line, joining their centres, passes perpendicularly through the point of impact. And
Oblique Percussion, is that in which the impulse is made in the direction of a line that does not pass through the common centre of gravity of the striking bodies; whether that line of direction is perpendicular to the place of impact, or not.
The force of Percussion is the same as the momentum, or quantity of motion, and is represented by the product arising from the mass or quantity of matter moved, multiplied by the velocity of its motion; and that without any regard to the time or duration of action; for its action is considered totally independent of time, or but as for an instant, or an infinitely small time.
This consideration will enable us to resolve a question that has been greatly canvassed among philosophers and mathematicians, viz, what is the relation between the force of Percussion and mere pressure or weight? For we hence infer, that the former force is insinitely, or incomparably, greater than the latter. For, let M denote any mass, body, or weight, having no motion or velocity, but simply its pressure; then will that pressure or force be denoted by M itself, if it be considered as acting for some certain finite assignable time; but, considered as a force of Percussion, that is, as acting but for an infinitely small time, its velocity being 0<*> or nothing, its percussive force will be 0 X M, that is 0, or nothing; and is therefore less than any the smallest percussive force whatever. Again, let us consider the two forces, viz, of Percussion and pressure, with respect to the effects they produce: Now the intensity of any force is very well measured and estimated by the effect it produces in a given time: But the effect of the pressure M, in 0 time, or an infinitely small time, is nothing at all; that is, it will not, in an infinitely small time, produce, for example, any motion, either in itself, or in any other body: its intensity therefore, as its effect, is insinitely less than any the smallest force of Percussion. It is true, indeed, that we see motion and other considerable effects produced by mere pressure, and to counteract which it will require the opposition of some considerable percussive force: but then it must be observed, that the former has been an infinitely longer time than the latter in producing its effect; and it is no wonder in mathematics that an infinite number of infinitely small quantities makes up a finite one. It has therefore only been for want of considering the circumstance of time, that any question could have arisen on this head. Hence the two forces are related to each other, only as a surface is to a solid or body: by the motion of the surface through an infinite number of points, or through a finite right line, a solid or body is generated: and by the action of the pressure for an infinite number of moments, or for some finite time, a quantity equal to a given percussive force is generated: but the surface itself is infinitely less than any solid, and the pressure infinitely less than any percussive force. This point may be easily illustrated by some familiar instances, which prove at least the enormous disproportion between the two forces, if not also their absolute incomparability. And first, the blow of a small hammer, upon the head of a nail, will drive the nail into a board; when it is hard to conceive any weight so great as will produce a like effect, i. e. that will sink the nail as far into the board, at least unless it is left to act for a very considerable time: and even after the greatest weight has been laid as a pressure on the head of the nail, and has sunk it as far as it can as to sense, by remaining for a long time there without producing any farther sensible effect; let the weight be removed from the head of the nail, and instead of it, let it be struck a small blow with a hammer, and the nail will immediately sink farther into the wood. Again, it is also well known, that a ship-carpenter, with a blow of his mallet, will drive a wedge in below the greatest ship whatever, lying aground, and so overcome her weight, and lift her up. Lastly, let us consider a man with a club to strike a small ball, upwards or in| any other direction; it is evident that the ball will acquire a certain determinate velocity by the blow, suppose that of 10 feet per second, or minute, or any other time whatever: now it is a law, universally allowed in the communication of motion, that when different bodies are struck with equal forces, the velocities communicated are reciprocally as the weights of the bodies that are struck; that is, that a double body, or weight, will acquire half the velocity from an equal blow; a body 10 times as great, one 10th of the velocity; a body 100 times as great, the 100th part of the velocity; a body a million times as great, the millionth part of the velocity; and so on without end: from whence it follows, that there is no body or weight, how great soever, but will acquire some sinite degree of velocity, and be overcome, by any given small finite blow, or Percussion.
It appears that Des Cartes, first of any, had some ideas of the laws of Percussion; though it must be acknowledged, in some cases perhaps wide of the truth. The first who gave the true laws of motion in nonelastic bodies, was Doctor Wallis, in the Philos. Trans. numb. 43, where he also shews the true cause of reflections in other bodies, and proves that they proceed from their elasticity. Not long after, the celebrated Sir Christopher Wren and Mr. Huygens imparted to the Royal Society the laws that are observed by perfectly elastic bodies, and gave exactly the same construction, though each was ignorant of what the other had done. And all those laws, thus published in the Philos. Trans. without demonstration, were afterwards demonstrated by Dr. Keill, in his Philos. Lect. in 1700; and they have since been followed by a multitude of other authors.
In Percussion, we distinguish at least three several sorts of bodies; the perfectly hard, the perfectly soft, and the perfectly elastic. The two former are considered as utterly void of elasticity; having no force to separate them, or throw them off from each other again, after collision; and therefore either remaining at rest, or else proceeding uniformly forward together as one body or mass of matter.
The laws of Percussion therefore to be considered, are of two kinds: those for elastic, and those for nonelastic bodies.
The one only general principle, for determining the motions of bodies from Percussion, and which belongs equally to both the sorts of bodies, i. e. both the elastic and nonelastic, is this: viz, that there exists in the bodies the same momentum, or quantity of motion, estimated in any one and the same direction, both before the stroke and after it. And this principle is the immediate result of the third law of nature or motion, that reaction is equal to action, and in a contrary direction; from whence it happens, that whatever motion is communicated to one body by the action of another, exactly the same motion doth this latter lose in the same direction, or exactly the same does the former communicate to the latter in the contrary direction.
From this general principle too it results, that no alteration takes place in the common centre of gravity of bodies by their actions upon one another; but that the said common centre of gravity perseveres in the same state, whether of rest or of uniform motion, both before and after the shock of the bodies.
Now, from either of these two laws, viz, that of the preservation of the same quantity of motion, in one and the same direction, and that of the preservation of the same state of the centre of gravity, both before and after the shock, all the circumstances of the motions of both the kinds of bodies after collision may be made out; in conjunction with their own peculiar and separate constitutions, namely, that of the one sort being elastic, and the other nonelastic.
The effects of these different constitutions, here alluded to, are these; that nonelastic bodies, on their shock, will adhere together, and either remain at rest, or else move together as one mass with a common velocity; or if elastic, they will separate after the shock with the very same relative velocity with which they met and shocked. The former of these consequences is evident, viz, that nonelastic bodies keep together as one mass after they meet; because there exists no power to separate them; and without a cause there can be no effect. And the latter consequence results immediately from the very definition and essence of elasticity itself, being a power always equal to the force of compression, or shock; and which restoring force therefore, acting the contrary way, will generate the same relative velocity between the bodies, or the same quantity of matter, as before the shock, and the same motion also of their common centre of gravity.
To apply now the general principle to the determination of the motions of bodies after their shock; let B and b be any two bodies, and V and v their respective velocities, estimated in the direction AD; which quantities V and v will be both positive if the bodies both move towards D, but one of them as v will be negative if the body b move towards A, and v will be = 0 if the body b be at rest. Hence then BV is the momentum of B towards D, and bv is the momentum of b towards D, whose sum is BV + bv, which is the whole quantity of motion in the direction AD, and which momentum must also be preserved after the shock.
Now if the bodies have no elasticity, they will move together as one mass B + b after they meet, with some common velocity, which call y, in the direction AD; therefore the momentum in that direction after the shock, being the product of the mass and velocity, will be . But the momenta, in the same direction, before and after the impact, are equal, that is ; from which equation any one of the quantities may be determined when the rest are given. So, if we would find the common velocity after the stroke, it will be , equal to the sum of the momenta divided by the sum of the bodies; which is also equal to the velocity of the common centre of gravity of the two bodies, both before and after the collision. The signs of the terms, in this value of y, will be all positive, as| above, when the bodies move both the same way AD; but one term bv must be made negative when the motion of b is the contrary way; and that term will be absent or nothing, when b is at rest, before the shock.
Again, for the case of elastic bodies, which will separate after the stroke, with certain velocities, x and z, viz, x the velocity of B, and z the velocity of b after the collision, both estimated in the direction AD, which quantities will be either positive, or negative, or nothing, according to the circumstances of the masses B and b, with those of their celerities before the stroke. Hence then Bx and bz are the separate momenta after the shock, and Bx + bz their sum, which must be equal to the sum BV + bv in the same direction before the stroke: also z - x is the relative velocity with which the bodies separate after the blow, and which must be equal to V - v the same with which they meet; or, which is the same thing, that ; that is, the sum of the two velocities of the one body, is equal to the sum of the velocities of the other, taken before and after the stroke; which is another notable theorem. Hencethen, for determining the two unknown quantities x and z, there are these two equations, viz, , and ; or ; the resolution of which equations gives those two velocities as below, viz, , and .
From these general values of the velocities, which are to be understood in the direction AD, any particular cases may easily be drawn. As, if the two bodies B and b be equal, then B - b = 0, and B + b = 2B, and the two velocities in that case become, after impulse, x = v, and z = V, the very same as they were before, but changed to the contrary bodies, i. e. the bodies have taken each other's velocity that it had before, and with the same sign also. So that, if the equal bodies were before both moving the same way, or towards D, they will do the same after, but with interchanged velocities. But if they before moved contrary ways, B towards D, and b towards A, they will rebound contrary ways, B back towards A, and b towards D, each with the other's velocity. And, lastly, if one body, as b, were at rest before the stroke, then the other B will be at rest after it, and b will go on with the motion that B had before. And thus may any other particular cases be deduced from the first general values of x and z.
We may now conclude this article with some remarks on these motions, and the mistakes of some authors concerning them. And first, we observe this striking difference between the motions that are communicated by elastic and by nonelastic bodies, viz, that a nonelastic body, by striking, communicates to the body it strikes, exactly its whole momentum; as is evident. But the stroke of an elastic body may either communicate its whole motion to the body it strikes, or it may communicate only a part of it, or it may even communicate more than it had. For, if the striking body remain at rest after the stroke, it has just lost all its motion, and therefore has communicated all it had; but if it still move forward in the same direction, it has still some motion left in that direction, and therefore has only communicated a part of what motion it had; and if the striking body rebound back, and move in the contrary direction, the other body has received not only the whole of the motion that the first had, but also as much more as the first has acquired in the contrary direction.
It has been denied by some authors, and in the Encyclopédie, that the same quantity of motion remains after the shock, as before it; and hence they seize an opportunity to reprehend the Cartesians for making that assertion, which they do, not only with respect to the case of two bodies, but also of all the bodies in the whole universe. And yet nothing is more true, if the motion be considered as estimated always in one and the same direction, esteeming that as negative, which is in the contrary or opposite direction. For it is a general law of nature, that no motion, nor force, can be generated, nor destroyed, nor changed, but by some cause which must produce an equal quantity in the opposite direction. And this being the case in one body, or two bodies, it must necessarily be the case in all bodies, and in the whole sola<*> system, since all bodies act upon one another. And hence also it is manifest, that the common centre of gravity of the whole solar system must always preserve its original condition, whether it be of rest or of uniform motion; since the state of that centre is not changed by the mutual actions of bodies upon one another, any more than their quantity of motion, in one and the same direction.
What may have led authors into the mistake above alluded to, which they bring no proof of, seems to be the discovery of M. Huygens, that the sums of the two products are equal, both before and after the shock, that are made by multiplying each body by the square of its velocity, viz, that , where V and v are the velocities before the shock, and x and z the velocities after it. Such an expression, namely the product of the mass by the square of the velocity, is called the vis viva, or living force; and hence it has been inferred that the whole vis viva before the shock, or BV2 + bv2, is equal to that after the stroke, or Bx2 + bz2; which is indeed very true, as will be shewn presently. But when they hence infer, both that therefore the forces of bodies in motion are as the squares of the velocities, and that there is not the same quantity of motion between the two striking bodies, both before and after the shock, they are grossly mistaken, and thereby shew that they are ignorant of the true derivation of the equation . For this equation is only a consequence of the very principle above laid down, and which is not acceded to by those authors, viz, that the quantity of motion is the same before and after the shock, or that , the truth of which last equation they deny, because they think the former one is true, never dreaming that they may be both true, and much less that the one is a consequence of the other, and derived from it; which however is now found to be the case, as is proved in this manner:
It has been shewn that the sum of the two momenta,| in the same direction, before and after the stroke, are equal, or that ; and also that the sum of the two velocities of the one body, is equal to the sum of those of the other, or that ; and it is now proposed to shew that from these two equations there results the third equation , or the equation of the living forces.
Now because , by transposition it is - ; which fhews that the difference between the two momenta of the one body, before and after the stroke, is equal to the difference between those of the other body; which is another notable theorem. But now, to derive the equation of the vis viva, set down the two foregoing equations, and multiply them together, so shall the products give the said equation required; thus Mult. , the equat. of the momenta, by , the equat. of the velocities, produc. , or , the very equation of the vis viva required. Which was to be proved.
When the elasticity of the bodies is not perfect, but only partially so, as is the case with all the bodies we know of, the determination of the motions after collision may be determined in a similar manner. See Keill's Lect. Philos. lect. 14, theor. 29, at the end. And for the geometrical determinations after impact, see the article Collision.
Centre of Percussion, is the point in which the shock or impulse of a body which strikes another is the greatest that it can be. See Cenfre.
The Centre of Percussion is the same as the centre of oscillation, when the striking body moves round a fixed axis. See Oscillation.
But if all the parts of the striking body move with a parallel motion, and with the same velocity, then the Centre of Percussion is the same as the centre of gravity.
