188 Dynamics. 



to this, it would only be necessary to substitute, in the express- 

 ions 2 u' u and 2 u' db D, the value of u' furnished by the 

 rules of articles 286, 288. But as these formulas would not pre- 

 sent themselves in a manner so easy to be retained as the rules 

 we have given, we leave this substitution to be made by those 

 who may wish to see the result. 



299. We observe that, when one of the two bodies is at rest, 

 the velocity which it would receive by the collision is double 

 that which it would have had, considered as non-elastic. This 

 is an evident consequence of the general rule. 



300. To give a few examples of these rules, let us suppose, in 

 the first place, that the two bodies are equal, and that one of 



them is at rest ; then , which expresses the velocity after 



wi -f n' 



collision, the bodies being considered as unelastic, becomes 

 or | u. From twice \ u or w, therefore, we subtract u to 



297. obtain the velocity of the impinging body after collision, which 

 is consequently zero. To find the velocity of the impinged body, 

 from twice ^ u or n, we subtract 0, which it had before collision, 

 and we have u for the velocity after collision. Hence we see 

 that the motion of the impinging body passes wholly into the 

 impinged. Accordingly, if several equal elastic bodies be placed 

 in contact with each other in the same straight line, and one of 

 them be made to impinge against the others in the direction of 

 this line ; the only effect would be, that the one at the opposite 

 extremity would be driven off with the same velocity. If two 

 are made to impinge at the same time against the others, two 

 would be detached from the other extremity, and so on. 



Let us suppose the two bodies to move in the same direction, 

 one of 5 ounces, and with a velocity of 6 feet in a second, and 

 the other of 7 ounces, with a velocity of 2 feet in a second. For 

 the common velocity which they would have after collision, con- 

 sidered as unelastic, we obtain 



5 XG -f 7 X 2_ 44 _ 



-~5TT~ 



If, therefore, from double this quantity or 7 1, we take the veloc- 

 ities before collision, namely, 6 and 2 respectively, we shall have 



