720 MECHANICS Impact 181. When two balls of glass or ivory impinge on one of small another, the portions of the surfaces immediately in contact spheres. are di 8 fig ure( i anc [ compressed until the molecular reactions thus called into play are sufficient to resist further distortion and compression. At this instant it is evident that the points in contact are moving with the same velocity. But as solids in general possess a certain degree of elasticity both of form and of volume, the balls tend to recover their spherical form, and an additional impulse is generated. This is proportional, as Newton found by experiment, to that exerted during the compression, provided neither of the bodies is permanently distorted. The coefficient of proportionality is a quantity determinable by experiment, and may be conveniently termed the " coefficient of resti tution." It is always less than unity. 182. The method of treating questions involving actions of this nature will be best explained by taking as an example the case of direct impact of one spherical ball on another. It is evident that in the case of direct impact of smooth or non-rotating spheres we may consider them as mere particles, since everything is symmetrical about the line joining their centres. If the impinging masses are of large dimensions, of the size of the earth, for instance, we cannot treat the effects of the impact independently of the other forces involved ; for the duration of collision in such a case may be one of hours instead of fractions of a second. 183. Suppose that a sphere of mass M, moving with a speed v, overtakes and impinges on another of mass M , moving in the same straight line with speed v, and that, at the instant when the mutual compression is completed, the spheres are moving with a common speed V, Let R be the impulse during the compression, then whence i + MV M + M ,
MM . . . (I). From these results we see that the whole momentum after impact is the same as before, and that the common speed is that of the centre of inertia before impact. The quantity V can vanish only if that is, if the momenta were originally equal and opposite. This is the complete solution of the problem if the balls be inelastic, or have no tendency to recover their original form after compression. 184. If the balls be elastic, there will be generated, by their tendency to recover their original forms, an addi tional impulse proportional to R,. Let & be the coefficient of restitution, and v v v^ the speeds of the balls when finally separated. Then, as before, =cB, whence and M + M -- -M+V 1 - ^-^ 1 with a similar expression for v^. These results may be more easily obtained by the simple consideration that the whole impulse is (1 + e)~R ; for this gives at once M(v-t j) = M (y i - v ) = (1 + e)R. If M be inBnite, and v = 0, we have the result of direct impact on a, fixed surface, viz., v - i = (1 + e)v or v l = - ev. The ball rebounds from the fixed surface with a speed e times that with which it impinged. 185. Suppose, now, M = M , e = 1 ; that is, let the balls be of equal mass, and their coefficient of restitution unity (or, in the usual but most misleading phraseology, suppose the balls to be " perfectly elastic"); then 2R = M(v - v ) ; v l v, and similarly v^ = v; or the balls, whatever be their speeds, interchange them, and the motion is the same as if they had passed through one another without exerting any mutual action whatever. Thus if a number of equal solitaire balls or billiard balls be arranged in contact in a horizontal groove, and another equal ball impinge on one extremity of the row, it is re duced to rest, and the ball at the other end of the row goes off with the original speed of impact. If two im pinge, two go off, and so on. 186. We may write the above expressions in terms of the impulse, thus R(i+c) , V ^ V ~ M i . (2). Consei- vation mome: tuni. Hence Mvj + M f/ Mv + MV, whatever e be, or there is no momentum lost. This is, of course, a direct con sequence of the third law of motion. Again plj + pIVJ = pi;; 2 + pi V s c}(v - v ) + P 2 (l + *)*^r M + M MM The last term of the right hand side is therefore the Loss of kinetic energy apparently destroyed by the impact. When ener sy. e = 0, its magnitude is greatest, and equal to When e=l,its magnitude is zero; that is, when the coefficient of restitution is unity no kinetic energy is lost. The kinetic energy which appears to be destroyed in any of these cases is, as we see from 171, only transformed partly it may be into heat, partly into sonorous vibrations, as in the impact of a hammer on a bell. But, in spite of this, the elasticity may be " perfect." Hence the absurdity of the designation alluded to in 185, Also by (2) = c(v-v ) by (1). Hence the velocity of separation is e times that of approach. 187. Two smooth spheres, moving in given paths and Obliqm with given speeds, impinge ; to determine the impulse and ini r act - the subsequent motion, Let the masses of the spheres be M, M , their speeds before impact v and v , and let the original directions of motion make with the line which joins the centres at the instant of impact the angles a, a , which may be calculated from the data, if the radii of the spheres be given. Since the spheres are smooth, the entire impulse takes place in the line joining the centres at the instant of impact, and the future motion of each sphere will be in the plane passing through this line and its original direction of motion. Let R be the impulse, e the coefficient of restitution ; then, since the speeds in the line of impact are vcosa and i/cosa , we have for their final values v v v/, after restitution, by 184 the expressions v, =rcosa- ~- ^-.(1 +c)(vcosa- t cosa ) , M + M M v^ = i/ cos a + and the value of R is M + M V c](vcosa- v cosa), (1 +c}(v cos a -v cosa) . M + M
Hence, the sphere M has finally a speed i in the lino