1 6.] IMPACT OF SPHERES. 7 



whence mv mu=(m'v' m'u'), 



or mv + m'v 1 = mti+m'u* ; (7) 



that is, the total momentum after impact is equal to that before 

 impact. 



14. This proposition will evidently hold for any number of 

 spheres whose centres move in the same line, and can then 

 be expressed in the form 



^mv ^mu. (8) 



It can be regarded as a special case of the so-called principle 

 of the conservation of the motion of the centroid to be proved 

 hereafter for any system not acted upon by external forces. 

 On the other hand, the proposition can be looked upon as a 

 further generalization of Newton's first law of motion. While 

 the latter asserts that the momentum of a particle remains 

 unchanged as long as no external forces act upon it, our law of 

 impact asserts the same thing for the momentum of a system. 



15. If the spheres were perfectly non-elastic, there would be 

 only compression and no subsequent extension. As at the end 

 of the period of compression, the velocities u, u' have both 

 become equal, viz. =w (Art. 12), the spheres after impact 

 would have the common velocity 



/ x 

 (9) 



m + m' 



. / 



16. If the spheres were perfectly elastic, i.e. if the elastic stress 

 following the compression, or the so-called force of restitution, 

 were just equal to the preceding stress of compression, the 

 spheres would completely regain their original shape. In this 

 case, the elastic stress causes the impinging sphere m to lose 

 during the period of restitution an amount of momentum 

 m(w u} equal to that lost during the period of compression. 

 Hence, the final velocity of m after impact would be 



