190 Dynamics** 



When, on the other hand, the bodies move before collision 

 in opposite directions, their velocities after collision are, 



2 u' u and 2 ' -J- *>, 



the difference of which is u + i>, and this was their relative velo- 

 city, or that with which they approached each other before col- 

 lision. Therefore the velocity with which they separate from 

 each other after collision, is the same as that with which they 

 approach each other before collision ; thus, with respect to elastic 

 bodies the relative velocity is the same before and after collision. 



Of the Motion of Projectiles. 



302. By the motion of projectiles, we understand that of 

 bodies, which, being thrown with a certain force, are afterward 

 left to the action of this force and that of gravity. We shall first 

 seek the path that would be described in free space. 



p. , 49 From the point A, let a body be thrown in the direction AZ, 

 and with any given velocity. If gravity were out of the ques- 

 tion, it would move uniformly in the direction of the straight line 

 AZ. But as gravity acts without interruption, the body will not: 

 be in the straight line AZ, except for an instant ; instead of AZ, 

 it will describe a curved line ABC of which AZ will be the tan- 

 gent at the point A. since AZ is one of the instantaneous direc- 



Geom. * . , -11 



97. tions of the moving body. 



303. In order to determine the nature of this curve, let AE 

 be the velocity communicated to the projectile, or the number of 

 feet that it would describe in a second, if it preserved continual- 

 ly this velocity ; and at the instant of its leaving the point A, let 

 us suppose this velocity composed of two others one AD hori- 

 zontal, and the other AF in a vertical direction. It is evident 

 that the direction of gravity being vertical or perpendicular to 

 AD, its action will not tend either to diminish or increase the 

 velocity AD, and that consequently whatever course the body 

 may take, it will preserve constantly the same velocity parallel 

 to the horizon. 



As to the velocity in the direction AF, when the body, in 

 virtue of its constant velocity, parallel to the horizon, shall have 



