258 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



spheres. The experiments were conducted with the aid of the 

 ballistic balance (Article 93), and the spheres came into contact 

 when moving in the line joining their centres. He found that the 

 relative velocity of the spheres after impact was oppositely directed 

 to that before impact, and that its amount was diminished in a 

 ratio depending only on the materials. 



To express this result, let u and u be the velocities of the two 

 bodies in the line of centres and in the same sense before impact, 

 v and v their velocities in the same line and in the same sense 

 after impact, then 



v v e (u u ) (1) 



where e is a positive fractional coefficient depending on the 

 materials of the two bodies, and independent of their masses 

 and velocities. 



If m and m are the masses of the bodies, we have by the 

 Principle of the Conservation of Linear Momentum (Article 111) 



mv 4- m v = mu + m u (2). 



Equations (1) and (2) determine the velocities after impact and 

 we find 



_ (m em } u + m? (1 + e) u 



m+m 

 , _ (mf em) u + m (1 + e) u 



m + m 



The impulsive pressure exerted by m on m is m (u v), in the 

 sense opposite to that of u, and this is 



(1 + e) mm (u u )/(m + m). 



The hypothesis adopted by Poisson and others as to the action 

 between two bodies in collision is founded on the result last 

 obtained. Poisson imagined that when the bodies come into 

 contact they begin to be compressed, that the compression con 

 tinues to increase until the end of a certain interval, called 

 the &quot;period of compression,&quot; and that at the end of this in 

 terval the two bodies have the same velocity along the common 

 normal to the surfaces in contact. Further, he imagined that 

 after the period of compression the bodies begin to recover their 

 form and continue to do so until they cease to be in contact. 

 The interval in which the recovery takes place is called the 

 &quot;period of restitution.&quot; Poisson supposed that the pressure 



