IMPACT OK COLLISION.] MECHANICAL PHILOSOPHY. DYNAMICS. 



745 



M'R 2 MRr 



for the accelerating force of the descending weight W. 

 The velocity, in t seconds, of the former, is 



v=ft 

 and the space, s = /t 2 = 



M'Rr Mr 2 

 M'fir4*r- s9 * 



M'Rr Mr* 

 M 7 



And for the latter we have 



M'R 2 MRr 



r -'"~ ~ ;/ 



M'R' -MRr 

 * "M"' R 4 Mr* J 



The student will perceive that the first member of the 

 equation (1), expressing the difference between the 

 moving forces that would act if the weights were free, is 

 the whole amount of moving force in actual operation ; 

 which whole amount is obviously expressed by the second 

 member of the equation. 



IMPACT OR IMPULSE. In this article we propose 

 to consider the circumstances of the motions of bodies 

 moving in certain straight lines from the effects of im- 

 pulse, and impinging against one another. 



Let any specified quantity of matter be considered as 

 unit of mass : if it be projected by any given im- 

 pulse, it will move with a constant velocity proportional 

 to the intensity of that impulse ; this velocity may 

 therefore be taken to represent the magnitude of the 

 impulsion. If we take two such units of mass, and two 

 such impulses act on them simultaneously, and in the 

 same direction, the relative positions of the moving 

 masses will be always preserved ; so that if the two 

 units be blended into one mass, and the two impulses be 

 thus made to unite and form a double impulse, the same 

 velocity will be impressed on the compounded mass. 

 And it is plain that xf M such units be blended into one 

 mass, which receives an impulse M times the intensity 

 we have supposed to be applied to the single unit, the 

 same velocity would still be impressed. 



Consequently, when any body whose mass Is M moves 

 from the effect of an impulse, the correct expression for 

 the intensity of that impulse must be Mu = the mass 

 multiplied by the velocity, and which, as before stated 

 (p. 743), is the momentum of the mass : hence the mo- 

 mentum measures the intensity of the impulse. 



COLLISION OF BODIES : DIRECT IMPACT. 

 When two bodies moving by impulsion in the same 

 straight line come into collision, the shock is called direct 

 impact. Suppose the bodies which thus impinge to be 

 entirely inelastic, or of such materials that they are 

 blended by the impact into one mass, our object will be, 

 from knowing the intensity of each separate impulse 

 that is, the momentum of each body to determine the 

 momentum of the blended mass ; and thence the cir- 

 cumstances of its motion. The following example will 

 illustrate how the necessary particulars are to be ascer- 

 tained : 



Example. Two inelastic bodies, whose masses are 

 M, M', move in the same straight line, with velocities 

 c, t/ : required the velocity of their united masses after 

 impact. 



The momentum of M is Mr, and that of M', MV, the 

 combination of these namely, Me 4 MV, is the mo- 

 mentum of the united mass M 4 M'. 



But if v be the velocity of this mass, its momentum 

 must be (M 4 M*) "i : hence, to determine the velocity 

 c, after the collision, we have the equation 



(M4M>, =M4MV. 

 MH-MV .(i), 



tCl M4M 



the velocity required. 



This result is on the supposition that the bodies are 

 moving in the same direction, or that one of them, M', 

 overtakes the other ; but if they are moving in opposite 



VOL. I. 



directions so that the two meet, then the expression for 

 t>j will be 



Me co MV /o\ 



*(}. 



If MB exceed MV, then, after the collision, the mass 

 will move in the direction in which M was proceeding ; 

 but if MV exceed Me, it will move in the direction in 

 which M' was proceeding, as is obvious. 



If Mi- = MV, then the collision, at meeting, will de- 

 stroy the motion of each, and bring both bodies to a 

 stand-still, since V 1 will then be 0. 



If one of the masses M' be at rest, then since n, = 0, 

 the velocity after impact will be 



^ = J^U.....(3). 



In all cases, the momentum after impact must be the 

 sum of the momenta before impact, regarding those 

 which act in opposite directions as having opposite signs ; 

 so that the momentum lost by one body, by the collision, 

 is exactly equal to the momentum gained by the other. 

 In the case (3) above, the momentum of the whole mass, 

 after impact, being (M 4 M')"i ~ Me, and the mo- 

 mentum gained by M', which was at rest, being M't^, 

 this must be the momentum lost by M. This loss of 

 momentum in M shows that the mass M' opposes a re- 

 sistance to the communication of motion, and M'e, ex- 

 presses the value of that resistance, which, as the mass 

 was at rest, can be due only to the inertia of the body. 



Let us now suppose that the bodies, instead of being 

 perfectly inelastic, are perfectly elastic ; and that, as in 

 the former case, they move so as to impinge at some 

 point in the common line described by their centres of 

 gravity. 



Elastic bodies are such as yield to the force of impact, 

 and undergo a compression, and therefore a change of 

 figure : the elasticity is that inherent force which the 

 body exerts to recover its original form. 



If the restoring force which the body exerts to recover 

 its original figure be equal to the impressing force at the 

 point of impact, so that the original form of the body is 

 perfectly restored, and in the same length of time that 

 it took to alter that form, the elasticity is said to be 

 perfect. 



In the case of perfect elasticity, whatever velocity one 

 body loses during the action of the compressing force, it 

 afterwards loses just as much more during the action of 

 the restoring force ; and whatever velocity the other 

 body gains during the compression, it gains as much 

 more during the restitution. ; . For, at the end of the 

 time occupied in the compression, there is the same com- 

 munication of momentum S in the case of inelastic 

 bodies : at that instant the; bodies, then in the closest 

 union, must have equal velocities. The force of resti- 

 tution, exactly equal to that of compression, then acts ; 

 and another effect, the opposite to that of collision, takes 

 place : the collision brings the bodies together ; the force 

 of elasticity drives them asunder. 



No bodies in nature are perfectly elastic, or completely 

 inelastic. The most elastic substance at present known 

 is glass. Newton, by a 

 contrivance similar to that 

 represented in Fig. 150, de- 

 termined a very close approx- 

 imation to the defect from 

 perfect elasticity of several 

 substances. If two balls of 

 the same substance, and in 

 every other respect equal, 

 were suspended from A by 

 slender threads, and each 

 let fall from equal distances, 

 measured from the vertical, 

 on the graduated arc B C, 

 they would, after collision 

 at o, each return to the 

 point in the arc from which it started, if the elasticity 

 were perfect. With ivory balls, the elasticity bore to 



5o 



Fig. 150. 



