8 KINETICS OF A PARTICLE. [17. 



Similarly, we have for the other sphere m* 



v' = w+(w u') = 2 w ti'. 



As w is known from (9), the velocities after impact can be 

 determined by means of these formulae for perfectly elastic 

 spheres. 



17. In general, physical bodies are imperfectly elastic, the force 

 of restitution being less than that of the original compression ; 



that is, we have 



(w v) = e(u w), 



(v' w)= e(w u r ), 



where e is a proper fraction whose limiting values are o for 

 perfectly inelastic bodies and i for perfectly elastic bodies. 

 This fraction e, whose value for different materials must be 

 determined experimentally, is called the coefficient of restitution 

 (or less properly, the coefficient of elasticity). 



18. To eliminate w we have only to add the last two equa- 

 tions ; this gives 



v 1 v = e(u u')' y (10) 



that is, the ratio of the relative velocity after impact to the rela- 

 tive velocity before impact is constant and equal to the coefficient 

 of restitution. 



This proposition, in connection with the proposition of Art. 

 13, expressed by formula (7), is sufficient to solve all problems 

 of so-called direct impact, i.e. when the centres of the spheres 

 move in the same line. 



19. As the coefficient e is frequently difficult to determine, 

 the limiting cases e=o, e=i are important as giving approxi- 

 mate solutions for certain classes of substances. 



Thus, for nearly inelastic bodies (such as clay, lead, etc.) 

 we may put e=o, whence, by (10), v 1 = v, i.e. the velocities of 

 the spheres become equal after impact ; and the value of the 

 common velocity is found from (7) as 



mu-\-m'u f 



v= 



