﻿ifon-JS'e wton la n Mechanics. 3 7 7 



relations u x = ^ — — -rs and u 2 — ^— — o. Since these velo- 



eomposition of velocities we find for these velocities the 

 u~v , —u — v 



i t~v and ?<2 — ~i ', r~2 # 



l—uv/c 2 1 + uv/c 1 



cities are not ot the same magnitude, the two bodies which 

 have the same mass when at rest do not now have the same 

 mass to this observer. Let us call these masses before collision 

 m { and m 2 . During collision, the velocities of the bodies will 

 all the time be changing ; from the principle of the conser- 

 vation of mass, however, the sum of the two masses will 

 always equal ?>? 1 + ?» 2 ~ x ". When in the course of the collision 

 the bodies have come to relative rest and are both moving 

 past our observer with the velocity —v, their momentum will 

 be — (m 1 + m 2 )v, and from the principle of the conservation 

 of momentum this must be equal to the original momentum 

 before collision, giving us the equation, — 



u — V — ?/ — V 

 — {m 1 + m^)v — m l u l + mc !i u 2 = m l - - +m 2 — . . (1) 



c 2 c 2 



Simplify inc:, we have,- 



m x 



= 



1- 



uv 



m 2 



1 + 



uv 



(2) 





, — U — V\ 2 / 



( , uv A 2 



\A-?' 



" H \/i- 



, U — V v 2 „ 



\A-5 



which by direct algebraic transformations may be shown 

 to be identical with 



(3) 



* In this connexion an interesting- fact has been pointed out to the 

 •writer by Professor Lewis. As stated above, the sum of the two masses 

 is throughout collision always equal to m x ~\-m«, and hence also at the 

 time in the collision when the masses have come to relative rest their 

 sum is w*] + m 2 . Since at this time both bodies are moving- with the 

 velocity — v we might suppose that m x + m 2 equals 2m / Vl —v-fc*. This 

 is not the case, however, since the bodies now possess additional elastic 

 energy beyond that which they possess when at rest and not in contact. 

 A relation between mass and energy has already been developed (ioec. ei#.), 

 and the mass of this elastic energy must also be taken into account in 

 calculating- )», + )»,, In fact the consideration of a collision of this type 

 leads to a simple proof of the relation between mass ami energ'V, a proof 

 presented by Professor Lewis in a series of lectures on tin* theory of 

 Relativity given at Harvard University in the Spring of I'M L. 



