A(1+e) u then the second acquires the velocity A+B · Suppose that the second body now impinges directly with this velocity on the third at rest; then the third acquires the B(1+e) 4(1+e) AB (1+e) u Now (A+B)(B+C)* supposing every quantity given except B, required to determine B so that the velocity communicated to C may velocity B+C A+B u, that is be the greatest possible. We must therefore make the denominator of the last fraction as small as possible. AC B+A+C+ B But B+A+C+ AC = (√B- √ AC)2 + (NA + NO), 2 B so that the least value is that when B-vanishes, that is when B=√(AC). Hence the velocity communicated to the third body is greatest when the mass of the second body is a mean proportional between the masses of the first and third. 110. The theory of the collision of bodies appears to be chiefly due to Newton, who made some experiments and recorded the results: see the Scholium to the Laws of Motion in Book I. of the Principia. In Newton's experiments however the two bodies seem always to have been formed of the same substance. He found that the value of e for balls of worsted was about for balls of steel 5 9' about the same, for balls of cork a little less, for balls of An extensive series of experiments was made by Mr Hodgkinson, and the results are recorded in the Report of the British Association for 1834. These experiments shew that the theory may be received as satisfactory, with the exception that the value of e, instead of being quite constant, diminishes when the velocities are made very large. EXAMPLES. IX. 1. An inelastic body impinges on another of twice its mass at rest: shew that the impinging body loses twothirds of its velocity by the impact. 2. A body weighing 5 lbs. moving with a velocity of 14 feet per second, impinges on a body weighing 3 lbs., and moving with a velocity of 8 feet per second: find the velocities after impact supposing e= 1 3 3. Two bodies are moving in the same direction with the velocities 7 and 5; and after impact their velocities are 5 and 6: find the index of elasticity. 4. Two bodies of unequal masses moving in opposite directions with momenta numerically equal meet: shew that the momenta are numerically equal after impact. 5. A body weighing two lbs. impinges on a body weighing one lb.; the index of elasticity is: shew that 2v=u+u', and that v'=u. 6. The result of an impact between two bodies moving with numerically equal velocities in opposite directions is that one of them turns back with its original velocity, and the other follows it with half that velocity: shew that one body is four times as heavy as the other, and that e= 1 7. Find the necessary and sufficient condition in order that may be equal to u. 8. A strikes B, which is at rest, and after impact the velocities are numerically equal: if r be the ratio of B's mass to A's mass, shew that the index of elasticity is and that B's mass is at least three times A's mass. 9. A, B and C are the masses of three bodies, the first and third of which are formed of the same substance; the first impinges on the second at rest, and then the second impinges on the third at rest: determine the index of elasticity in order that the velocity communicated to C may be the same as if A impinged directly on C. 10. A body impinges on an equal body at rest: shew that the vis viva before impact cannot be greater than twice the vis viva of the system after impact. 11. A series of perfectly elastic bodies are arranged in the same straight line; one of them impinges on the next, then this on the next, and so on: shew that if their masses form a Geometrical Progression of which the common ratio is r, their velocities after impact form a Geometrical Progression of which the common ratio is 2 r+1 12. A number of bodies A, B, C,... formed of the same substance, are placed in a straight line at rest. A is then projected with a given velocity so as to impinge on B; then B impinges on C; and so on. Find the masses of the bodies B, C,... so that each of the bodies A, B, C,... may be at rest after impinging on the next; and find the velocity of the nth ball just after it has been struck by the (n-1)th ball. X. The Oblique Collision of Bodies. 111. In the present Chapter we shall consider the oblique collision of bodies; see Art. 97. It will be found that the problems discussed involve only a more extensive application of principles already explained. We shall confine ourselves to cases in which the line of impact and the directions of the motions of the bodies are in one plane. 112. A body impinges obliquely on another: required to determine the velocities after impact, the elasticity being imperfect. Let a body whose mass is m, moving with a velocity u, impinge on another whose mass is m', moving with a velocity u. Let the direction of the first velocity make an angle a with the line of impact, and that of the second an angle a. After impact let the velocities be denoted by v and v', and the angles which their directions make with the line of impact by ẞ and B'. Resolve all the velocities along the line of impact and at right angles to it. No impulsive force acts on the bodies in the direction at right angles to the line of impact, and so the velocities at right angles to the line of impact remain unchanged. Hence The velocities along the line of impact are affected just as they would be if the velocities in the other direction did not exist. Hence, proceeding as in Art. 101, we obtain mu cos a+m'u' cos a' — em' (u cos a — u' cos a') v cos B 'cos B' = m+m' mu cos a+m'u' cos a' + em (u cos a—u'cos a') ....(3), ....(4). If we divide (1) by (3) we obtain the value of tan ß; this determines the direction of the velocity of the impinging body after impact. If we square (1) and (3) and add, we obtain the value of v2; this determines the magnitude of the velocity. Similarly from (2) and (4) we can determine the direction and the magnitude of the velocity of the other body after impact. sents the centre of the impinging body, considered to be a sphere, at the instant of impact; and C' the centre of the other body. CC is the line of impact. The directions of the velocity of the imping B D ing body before and after impact are represented by CA and CB; and those of the other body by C'A' and C'B'. Thus if D be a point on CC' produced, angle BCD=ß, angle B'C'D=ß'. This figure may serve to illustrate the problem; it will however be easily perceived that the general formulæ admit of application to a large number of special cases, and that the figure would have to be modified in order to apply accurately to such special cases. For instance, we have supposed CA and CA' to fall on the same side of CD, but it is of course possible that they should fall on different sides. 114. Multiply equation (3) of Art. 112 by m, and equation (4) by m' and add; thus we obtain mv cos ß+m'v' cos B'= this shews that the momentum of the system resolved |