ol.USION. IMPACT. 



COLLISION, IMPACT. 



Now. suppose U ball A (which U so null that iU aiie may be 

 Abated) to approach obliquely toward* the obstacle XT. say in thr 

 JheeUuu Ok Let CD be UM velocity, or length moved over in one 

 [VELOCITY] the velocity CD U equivalent to the two 



CK and KD. The 6rat U destroyed, and then partially re- 

 by the impact ; the second remains unaltered, except by the 

 friction at the moment of impact, which we do not consider. If then 

 we take DL equal to KD, and draw LM perpendicular to XT, and in 

 length uch a fraction of KC ai e is of I, the ball will move after im- 

 pact with the velocities DL and LM, that is, with the velocity DM in the 

 direction DM. If the system were perfectly inelastic, the ball would 

 proceed along DL ; if perfectly elaatic, ML would be equal to CK, and 

 DM and CD equally inclined to XT. If the size of the ball be taken 

 into account, XT must be nippoeed to be a line parallel to the obstacle, 

 and distant from it by the radius of the ball. 



Direct impart tout nUitiom. Let the nnssm of the two balk or 

 material particle* be m and m', and let them move with uniform 

 velocities r and r* in the same direction along a straight line, r being 

 greater than r 1 , so that m overtakes m'. Let be the common velocity 

 of the two balls when the compression at the moment of impact in a 

 maximum ; let r be the momentum spent in producing this com- 

 pression, and r the momentum acquired during the restitution of the 

 form of the bodies, being the coefficient of elasticity. Let v and r' 

 be the velocities of the balls when collision ceases. Hence, we have 

 the three following 



(1.) m r = momentum of M at the beginning of collision. 



F = momentum spent in producing compression. 

 m* = momentum of m, when compression is a mar. 



(2.) m't'= momentum of ' at the beginning of collision. 

 I'M = momentum of m', when compression is a max. 



.-. mV='-p. 

 (3.) At the instant when collision ceases, we have similarly, 



m v =w u t p. 



from which equations we shall get 



M = 



..' 



'v" 



rt + in' 



m+m' 



- 



w-t-m' 



wn-m' 



_ 

 +' 



Ottioue impact and atiuum. In oblique ini|ct we assume that the 

 mutual action of the balls during collision JH along the lino joining 

 their centres at the instant when compression is a maximum, and along 

 that line only ; that is, we assume the balls to be perfectly smooth. 

 Hence, if a smooth ball impinges obliquely on a smooth plane, the line 

 of reaction of the plane will be perpendicular to its surface, and the 

 momentum of the impinging ball will be affected along that line 



away to M ; let 

 which are there- 



in fy. 1 , let the ball c impinge at D, and be carried 

 * and v' be the velocities before and after impact, and 

 fore proportioned to CD and DM. 



CD 



DM" 



CF 

 ME* 



-in >' 



nine 



andt = !i = l. E = 



DF ME DF 



or 



ME 



tan 6' 

 tan 



The equations for the impact of two smooth baUs arc himil.-irly 

 fanned, and may be easily understood from their analogy to those 

 given abore for rfi'rert impact Thus, suppose two balls A and B to 



, an B o 



fai directions Mque to one another, and to strike each other. 

 Decompose the velocity of each ball into two, one in the line joining 

 the centre* at the moment of impact, and the <.th<-r iK-rpendiculnr to 

 The pair of velocities perpendicular to the central line will not be 

 altered by the impact ; and as tar as the remaining velocities are con- 

 cerned, the case is precisely the one already solved. Find the velocities 



in the central line as altered by the impact, compound them with the 

 perpendicular velocities which remain unaltered, and the resulting 

 velocities and directions will be those with which the bull* will pro- 

 ceed after the impact 



To take the most simple case, suppose the ball A, moving in the 

 direction EC, and with the velocity EC, to strike the ball B, which is at 

 rest Join D and c, the oxntres of the balls, and decompose BO into re 

 in the line joining the centres, and IF perpendicular to it Then A 



Fit. I. 



will only strike the ball B with the velocity FC. Suppose that l>y the 

 preceding rules it is found that A, striking B at rest with the velocity 

 FC, will be thrown back with the velocity Co, while B is struck forward 

 with the velocity UK. Then B will receive this velocity, and this one 

 only ; as to A it has after the impact acquired the velocity (;, which it 

 combines with the velocity OL, equal and parallel to KF ; so that 

 CL represents the velocity and the direction of the motion of A after 

 the impact 



In every case of impact, when the balls approach one another with 

 uniform velocities, the centre of gravity of the balls moves uniformly, 

 and in a straight line. After the impact, though the direction 

 velocities of the balls may have changed, yet their centre of gravity 

 still continues to describe the same line, and with the same velocity 

 as before. This proposition is proved in all works on elementary 

 mechanics. 



The following conclusions will now be readily deduced by any one 

 who understands the preceding results : 



1. If two inelastic balls move in the same direction, they <]o not 

 separate after the impact, but either move on with a common velocity, 

 or are reduced to rest If both move in the same direction, tin- 

 velocity after impact is (Art + BA)-t-(A + B) ; but if they move in different 

 directions, the motion after impact is in the direction of that Kill of 

 which the momentum (AO or Bft) is the greatest, and the velocity i- 

 (AO B4)~ (A + B) or (fl4 AoH-(A + B). When the momenta are equal, 

 there is no motion after the impact If 4 = 0, or if one of the ball* l- 

 at rest before impact, the velocity after impact is A<H-(A + B). To 

 deduce these results, make = in the formulae, and give the \vi 

 their proper signs. 



2. If two perfectly elastic balls move in the same direction tln-v 

 separnte after the impact, the velocity of the foremost being augmented 

 from 6 to 6 + 2(a 4)A-r(A + B). But the velocity of the hindmost is 

 either retarded, altogether destroyed, or made to change its direction, 

 the algebraical formula for the velocity after impact being a 2(a 6) 

 B-f-(A + B). This is nothing when B exceeds A, and when 4 is to a as 

 B A is to 2n. And according as 4 is to a in a less or greater ratio 

 than the preceding ratio, A'B velocity is or is not altered in direct i. >n. 



If two perfectly elastic balls move in opposite directions, th.it i t 

 being called positive, the velocities of A and B after impact are deter- 

 mined in magnitude and direction by the formula] 



a (a + 4) and 4+(a + 4) . 



A+B A+B 



4. If two perfectly elastic balls be equal in magnitude, the velocity 

 of each after the impact is that which the other had before the impact, 

 1-olh in magnitude and direction. 



5. In all cases perfectly elastic balls recede from each other after 

 impact with the same velocity with which they approached tx-i'on- 

 ini|>act; since jf <=l,r w = o 4. But in every other case the rate 

 of recess after impact is the same proportion of the rate of aj>| 

 before impact which c is of 1. 



6. The rii rira, or product of the mass and njuare of the velocity, of 

 a couple of perfectly elastic balls is the same before and after impact ; 

 in every other case it U less after impact than before. 



For further mathematical developments and deductions from these 

 general formula:, we may refer especially to Professor Price's t \ 

 on ' Infinitesimal Calculus,' vol. iii. c. 8 and 10 ; to Goldinp i 

 ' Natural Philosophy,' p. 1 34, et icq. ; and to Professor Walker's t r 

 on ' Mechanics,' c. vii. p. 178. 



