FOR
by Virtue of which nothing is performed per/ahum. But it is needlefs to iniift further on this, fin ce the duration of any fhock may be determined from from the moft certain principles. There can be no fhock or collifion of bodies, without their making mutual impreffions on each other : thefe impreffions will be greater or lefs according as the bodies are more or lefs ifoft, other circumftances being the fame. In bodies called hard, the impreffions are fmall ; but a perfect hardnefs, which admits of no impreffion, feems inconfifteut with the laws of nature ' : Co that while the collifion lafts, the action of bo- dies is the refult of their mutually preffing each other. This preffure changes their ftatc ; and the forces exerted in percuffion are really preflures^ and truly vires mortita.; if we wii'i uie this expreffion, which is no longer proper, fince the pretended infinite difference between the vires vivte and mor- bus ceafes k , — [ ' See the article Hard Ladies, Suppl. k Eider-, ibid, p. 32, 33.]
The force of percuffion, refulting from the preftltres bodies exert on each other while the collifion lafts, may be perfectly known, if thefe preflures be determined for every inftant of the fhock.
The mutual action of the bodies begins the firft moment of their contact, and is then leaft ; after which this action in- creases, and becomes grea'teft when the reciprocal impreffions are ftrongeft 1 , If the bodies have no elafticity, and the im- preffions, they have received, remain, the forces will then ceafe. But if the bodies be elaftic, and the parts comprefled rcitore themfelves to their former ftate, then will the bodies continue to prefs each other, till they feparate. To compre- hend therefore perfectly the force of percuffion, it is requifite to" define, £Yft, the time the mock lafts, and then to affign the preffure eorrefponding to each inftant of this time : and as the effect ofr preflures in changing the ftate of any body may be known, we may thence come at the true caufe of the change of motion arifing from collifion. The force of percuffion, therefore, is no more than the operation of a va- riable preflilre during a given time; and to meafure this force we muft have regard to the time, and to the variations ac- cording to which the preilurc increales and decreafes. ['Vide infra.]
Mr. Euler has given us fomc calculations with refpect to thefe particulars '•". It will be fufHcient here only to illuftrate their tendency by the inftance he brings. Suppofe two bodies A and B ; that the hardnefs of thefe bodies is equal ; and fuch, that being prefled together with the force of loolb. the impreffion made on each is of the depth of T ^- b - part of a foot. Suppofe, farther, B to be at reft, and fixed, and that A ftrikes it with the velocity of 100 feet in a fecond ; ac- according to Mr. Euler, the ™xezte& force of compreffion will be equivalent to 400,0 lb. and this force will produce in each of thefe bodies an impreffion equal to 4s of a foot ; and the duration of the collifion, that is, till the bodies arrive at their greateft compreffion, will be about tois of a fecond. Mr. Eu- Jer, in his calculations, fuppofes the hardnefs of a body to be proportional to the force or preffure requifite to make a given impreffion on it ; fo that the force by which a given impreffion is made on a body, is in a compound ratio of the hardnefs of the body and of the quantity of the impreffion. But he obferves, that regard muft be had to the magnitude of the bodies, as the fame Impreffion cannot be made on the leaft bodies as on the greateft, from the defect, of fpace through which their component particles muft be driven : he, therefore, only confiders the leaft impreffions, and that the bodies are of fuch magnitudes, that with refpe£t to them the impref- fions may be looked upon as nothing. What he fuppofes about the hardnefs of bodies, neither implies elafticity nor its defect, elafticity only producing a reftitution of figure and impreffion when the {iTelnng force ceafes; but this reftitution needs not be here confidered. It is likewife fuppofed, that the bodies ihocking have plane and equal bafes, by which they touch each other in the collifion; fo that the impreffion hereby made diminishes the length of each body. It is far- ther to be cbferved, that in Mr. Euler's calculations, bodies are fuppofed fo constituted, that they may not only receive impreffions from the jorce s preffing them, but that a greater force is requifite to make a greater impreffion. This excludes all bodies fluid or folid in which the fame force may pene- trate farther and farther, providing it have time, without ever being in equilibria with the refiftance. Thus a body ma) continually penetrate farther into foftwax, although deforce impelling it be not incFeafed. In thefe, and the like cafes, nothing is required but to furmount the firft obftacles ; which being once done, and the connexion of parts broken, the pe- netrating body always advances, meeting with the fame ob- ftacles as before, and dcitroying them by an equal force. But Mr. Euler only confiders the firft obftacles which exift before any reparation-' of parts, and which are, no doubt, fuch, that a greater impreffion requires a greater force. This, indeed, principally takes place in elaftic bodies ; but it feems Jikewift to obtain in all bodies when the impreffions made on them art fmall, and that the contexture of their parts is not altered.
-[»Md. P . 37,/^]
Thefe things being prefnifed, let the mafs or weight of th< body A be cxprefled in general by A, and let its velocity be-
FOR
fore the fhock be that which it mieht acquire by felling from the height a. Farther, let the hardnefs of A be exp'refed by M, and mat of B by N, and let the amplitude of the bafe, by which the impreffion is made be cc; then will the greateft
compreffion be made with ifinij -^ X A '.a. There- fore if the hardnefs of the two bodies, and the plane of their contafl during the whole time of their collifion be the fame this fircewiB be ,/A a, that is, as the fquare root of the vis vvua of the fairing body A. And as </a Is proportional W the velocity of the body A, the force of percuffion will be in a compound ratio of the velocity and of the fubduplicate ra- tio or the mafs of the body faking : fo that in this cafe neither the Leibnit.an nor the Cartcfian proportions take place. But this force of percuffion depends chiefly on the hardnefs of the bodies; the greater this is, the greater will the force of per- cuffion be. If M=N, this force will be as •/Wc'cXAa, that is in a compound fubduplicate ratio of the vis viva of the body Unking, of the hardnefs, and of the plane of con- tact, tut if M, the hardnefs of one of the bodies, be infi- nite, the farce of percuffion will be as "SWcTXAa; at the fame time if M = N, this force will be as v r J ' N«XA «' Therefore all other things being equal, thefircc of percuffionj it the Unking bony be infinitely hard, will be to the lam of percuffion if both the bodies be equally haul, as / 2 to I Mr. Euler farther deduces from his calculation, that the im-
proffion received by the body A will be as / ?f *Af
V M+NxMa
and the impreffion on B will be as /.— Mj <A '! ]f
, . V M HNxNrc'
therefore, the hardnefs of A, that is M, be infinite, it wdl iurrer no impreffion, whereas that on B will extend to the depth of^ _£. But if the hardnefs of the two bodies be the fame, or that M = N, they will each receive equal im- preffions of the depth I —-. So that the impreffion re-
V 2rv cc ceiied by the body B, in this cafe;, will be to the impreffion it receives in the former as I to / 2 ". - ["Ibid. p. 46, 47.] Mr. Euler has likewife confidered and computed the cafe when the ftriking body has its anterior fin face convex, with which it ftrikes an immoveable body whole furface is plane °. He has alfo examined the cafe when both bodies are fuppofed immoveable p ; and from his formula? he deduces the known laws of the collifion of non-elaftic and elaftic bodies. He has alio determined the greateft prcfiures the bodies receive in thefe cafes s ; and likewife the impreffions made on them. In particular he fhews that the impreffion received by the bo- dy ftruck, or B, if moveable, is to the impreffion received by the fame body when immoveable as ./ B to v'A-J-B. — [ ° Ibid. p. 48. § xxiv. p Ibid. § xxv. xxvi. 1 1bid. S xxvii.J
This doclrine of Mr. Euler may ferve to fhew, that the dif- pute about the meafure of fines is very needlefs in phyfics ; fince the laws of motion may, independently of any hvpothe- fis about the meafure of the vis viva, be deduced from the moft uncontefted principles of preffure and time. But we doubt whether we fhall be enabled hereby to fettle the me- taphyfical part of this difpute to the fatisfaflion of both par- ties, each of which may afient to all that Mr. Euler fays, and yet adhere either to the Cartefians or Leibnitians.'— But whatever may be faid of the metaphyseal part of the difpute, it is certain, from experience, That the number of equal fprings requifite to produce any velocity in a given body, is always proportional to the i'quarc of the velocity to be produced. Thus if one fpring can, by- unbending half, produce one degree of velocity in a body, ic will require four equal fprings to produce two degrees of ve- locity in that body; nine fprings to produce three degrees of velocity, C5V. See the article Spring, Suppl Alfo, if a portion of a yielding fubftance, as clay, tallow, &c. be juft fufficient to flop the motion of a body moving with a certain degree of velocity, it will require four times the quantity of the fame refilling fubftance to flop it, if the velocity of the moving body be double, CSV. The fame holds in the refiftance of wood againft mufquet-balls, as Mr. Ro- bins obferves in his New Principles of Gunnery : fo that a ball moving with twice the velocity of another, will penetrate four times as deep into earth, clay, tallow, wood, &c. And in like manner if the action of one man, onehorfe, or other animal, can give a certain degree of velocity to a given mafs, it will require the action of four equal men, holies, C3V. to give the fame mafs two degrees of velocity ; nine men, horfes, Cifc. to give it three degrees of velocity ; and fo on. Thefe practical points have been put out of all qileftion by the experiments of Poleni, 'S Gravefande, Defaguliers, and others, and are of great ufe, although they do not decide the controverfy about the meafure of the force of bodies in mo- tion. "bRCE of inertia, or vis inertia. It may be a queftion, whe- ther the vis inertia of bodies can properly be called & force ? As
