Collision of elastic Bodies. 187 



lafter collision, considered as unelastic, u u' would be the 

 velocity lost by the .impinging body ; when, therefore, the recoil 

 of the spring (taking place in a direction opposite to the motion) 

 causes as much motion to be lost, as had already been lost by 

 the compression, there will remain only the velocity 



u 2 (u u') = 2 w' u. 



As to the impinged body, u' v is the velocity gained by col- 

 lision ; and we have seen that, by the recoil of the spring, it ac- 

 quires as much more ; it will have, therefore, 



v 4. 2 (u' v) = 2 u' v. 



This case comprehends that in which one of the two bodies is at 

 rest before collision* 



If the bodies move in opposite directions, the reasoning is 

 precisely the same for the one which has the greater quantity of 

 motion. As to the other, it would, considered as unelastic, 

 lose its velocity by the collision, and acquire another in the op- 

 posite direction ; u f being this velocity, we shall have v + u' 

 for the velocity lost. Doubling this effect, on account of the 

 bodies being elastic, and adding it to the original velocity r, 

 we have 



2 (o -f- u') v = 2 u' -f v. 



297. By attending to the resulting expression in each of the 

 above cases, it will be seen that the circumstances of the collis- 

 ion of bodies perfectly elastic are all comprehended in this single 

 rule ; 



Seek the common velocity which the two bodies would have after 

 collision, if they were destitute of elasticity, then from double this ve- 

 locity, take the velocity which each had before collision, and we shall 

 have the velocity of each after collision ; it being understood that 

 when the bodies move in opposite directions before collision, the 

 sign is to be given to the velocity of that body which has the 

 less quantity of motion. 



298. From the principles above laid down, we might easily 

 obtain, for the collision of elastic bodies, formulas which should 

 contain only the masses and velocities before collision. In order 



