

MECHANICAL PHILOSOPHY. DYNAMICS. 



{ELASTIC BODIES. 



perfect elasticity the ratio of Stofl; with glass balls, 



tho ratio was 15 to 1 recent experiment*, of a 



different kin .1. Mr i Igkinson has determined 



the ration ry -81, and for glass -94 j the 



.ring but very littlo from Uiat deduced by 



Newton. The fractional or decimal part that the elas- 



, in of perfect i-laiitieity in any substance, is caUod 



the modu/uj of elasticity of that substance. 



../. Two perfectly elastic bodies, whose masses 

 are M, M', moving with velocities t>, r", strike with direct 

 impact : it U required to determine their velocities after- 

 wards. Whil tho compression continues, the bodies 

 move as one mass ; and therefore with the velocity 



Mr + MV 



.(1) 



So that M will lose the velocity P-P,, and M' will 

 lose tho velocity t/ 1>, ; and these express the velocities 

 communicated, but in opposite directions, by the force 

 of restitution. Hence the velocities, after impact, will 

 be 



of the mass M, c-2(p-p,) . . : . (2) 

 and of the mass M', 1/-2 (P'-O.) ---- (3) 

 Or, substituting for P its value (1), we have 



velocity of M', p-2 



(5) 



If each mass be multiplied by the velocity it has after 

 impact, tho sum of the products will be M + MV ; 

 which is also the sum of the momenta before impact : 

 hence the sum of the momenta after impact is the same 

 as the sum of tho momenta before impact. If we sub- 

 tract (3) from (2), the remainder will bee 7 P: hence 

 the difference of the velocities is the same both before 

 and after impact. 



If M = M', that is, if the bodies are perfectly equal in 

 mass, as well as perfectly elastic, the expressions (4) and 

 -liow that, after impact, the bodies will exchange 

 their velocities, M' moving -with the velocity v, and M 

 with the velocity c'. Hence, if one of the bodies be 

 perfectly at rest, the other which strikes it will impart 

 to it its entire velocity, and will itself rest in its place. 



If the elasticity be imperfect, the foregoing formula: 

 will require modification. Let e be the modulus of 

 elasticity of the impinging bodies ; that is, let the force 

 of restitution after compression be only the eth part of 

 that of compression : then the additional velocity lost by 

 M from the action of this force, instead of being equal 

 to that lost directly namely, -,, will be only e 

 (v-v,); and tho additional velocity lost by M', only e 

 ('-,); so that the velocities after the impact will be 



of the mass M, v 



and of the 



M', '- 



-*,) 



-*,) 



M + M' 



Tho momenta after impact will be, for M, the first 

 of these expressions multiplied by M ; and for M', the 

 second multiplied by M': the original momenta are for 

 the former, Me, and for the latter MV. Hence, tho 

 momentum lost by one of the bodies, by the impact, is 

 gained by tho other : for, as the above expressions show, 

 this momentum is 



which are tho same in value, bnt opposite in signs. 



Wln-ii tho In -dies are perfectly elastic, it has been 

 - above, as an immediate consequence of equations 



(2) and (3), that tho sum of the momenta before im- 

 pact is the same as the sum of the momenta after im- 

 pact ; and also that the difference of tho .velocities of 

 the two bodies must be the sauie after impact as before ; 

 that is, calling the velocities after impact V, and V, wo 

 must have 



M V + M'V'-Me -f MV. 



Also V_V-t/ e. 



/.M(V )-M'(p' V), 

 and V + t>-'+V. 



Now, if we multiply those two last results together, 

 we shall have tho equation 



Consequently 



M (V* v*) - MV V"). 

 M V+ M'V-Mr'-f MV*. 



Kg. 151. 



We may therefore conclude that when the bodies are 

 perfectly elastic, the sum of the products of each body 

 into the square of its velocity, is the same both before 

 and after impact 



A particular name is given to the product of a mass 

 into the square of the velocity with which it moves it 

 is called the vis viva, or the living force ; so that in th 

 collision of perfectly elastic bodies there is no loss of vit 

 viva occasioned by the impact. The consideration of the 

 force, thus called the vis viva, enters largely into certain 

 inquiries connected with the motion of fluids. 



1. A ball whose elasticity is e, strikes a perfectly hard 

 plane the raised edge of 

 a billiard-table, for ex- 

 ample: required the motion 

 of the ball after the impact. 

 Let A B (Fig. 151) be the 

 straight line described by 

 the centre of the ball be- 

 fore impact, and call the 

 , angle A B D', a ; then, P 

 being the velocity along 

 ABC, let it be represented by BC. The components 

 of this velocity the one perpendicular to the plane, and 

 the other parallel to it are B D, BE. The component 

 B E is not modified by the impact ; but B D, by the 

 force of restitution, is converted into B D': hence, com- 

 pounding the velocities B E, B D', the path, and velocity 

 v , , after impact, will be denoted by B C'. 



The velocity BD'isxBD'=p cos. a, 

 The velocity B E is P sin. a ; 



because the velocity parallel to the plane is unaltered by 

 the impact, 



.'. p 1 s =(e cos. o) ! + (psin. a)*- *.(* cos.* a + sin* a) 

 =p s ( 2 cos. ! a + l oos. 1 a)=p J {l 



(l c 2 )c<- J } ... . (1) 

 For the direction B C', after impact, we have 



cot. o= 



D~C' 



CMll. .1 



(2) 



If the elasticity be perfect, that is, if e^l, \then, from 

 (1) and (2), c, =u ; and a=a : hence, in the case of per- 

 fect elasticity, the ball will rebound with the velocity 

 with which it struck ; and the path it takes will make an 

 angle of reflection equal to the angle of incidence. 



But if the elasticity be imperfect, that is, if e be less 

 than 1, then, from (1), v 1 will be leas than P ; and, from 

 (2), cot. a will be less than cot. n ; and consequently a 

 will bo greater than a : hence, in the case of imperfect 

 elasticity, the velocity after impact will be less than the 

 velocity before impact ; but the direction after impact 

 will make a greater angle with the perpendicular than 

 the direction before impact. 



2. The following diagram (Fig. 152) represents two 

 bodies, A and B, impinging upon each other at tho point o; 

 C D, representing in magnitude and direction the velo- 

 city of the former, and CD', in magnitude and direction, 



