1 7S Dynamics. 



Of the direct Collision ofundastic Bodies. 



286. Two unelastic bodies which meet, (or one of which falls 

 upon the other at rest) communicate, or lose, a part of their 

 motion ; and in whatever manner this takes place, we may al- 

 ways, at the instant of the collision, represent each body, accord- 

 ing to the principle of D'Alembert, as urged with two velocities, 

 one of which remains after the collision, while the other is des- 



133. troyed. 



287. Let us, in the first place, suppose ,the two bodies to 

 move in the same direction. That which goes the faster, will 

 evidently lose a part of its velocity by the collision, and the 

 other will gain by it. Let ra be the mass of the impinging body, 

 and u its velocity before collision ; n the mass of the impinged 

 body (which may be less or greater than m), and v its velocity 

 before collision. Let us suppose that the velocity u changes to 

 ' by the collision ; m will accordingly have lost u u'. I will 

 consider m as having, at the instant of collision, the velocity u', 

 and the velocity u u'. If we suppose, in like manner, that v 

 becomes T/ by the collision, n will have gained v' v ; I can 

 accordingly consider it, at the instant of collision, as having the 

 velocity ', in the direction of the actual motion, and the velocity 

 T/ T in the opposite direction ; since, on this supposition, it 

 will really have only the velocity i/ (i/ v) or v. 



As, therefore, among these four velocities there can, by sup- 

 position, remain only u' and t/, the two others u u', and 

 T/ T, must be destroyed in the act of collision. Now as these 

 are directly opposite, it is necessary that the quantities of mo- 

 tion^ which the bodies would have in virtue of these velocities, 

 should be equal ; we have, therefore, 

 33 ' m (u M') = n (v' v). 



Now in order that u' and v' may, as we have supposed, be 

 the velocities which the two bodies m and n have after col- 

 lision, these velocities must be such, that the impinging body 

 shall not have the greater action over the impinged, that is, 

 that the two bodies shall, after col'ision, proceed in company; 

 we have, accordingly, v' = u' ; and hence form the equation. 



