ACTION AND REACTION. 199 



If the masses A and B be equal, then their motions or velocities added to- 

 gether must be the motion of the united mass after impact, since no motion can 

 either be created or destroyed by that event. But as A and B move with a 

 common motion, this sum must be equally distributed between them, and 

 therefore each will move with a velocity equal to half the sum of their ve- 

 locities before the impact. Thus, if A have the velocity 7, and B have 5, the 

 velocity of the united mass after impact is 6, being the half of 12, the sum of 7 

 and 5. 



If A and B be not equal, suppose them divided into equal component parts, 

 and let A consist of 8, and B of 6, equal masses : let the velocity of A be 17, 

 so that, the motion of each of the 8 parts being 17, the motion of the whole will 

 be 136. In the same manner, let the velocity of B be 10, the motion of each 

 part being 10, the whole motion of the 6 parts will be 60. The sum of the 

 two motions, therefore, toward C is 196 ; and since none of this can be 

 lost by the impact, nor any motion added to it, this must also be the whole 

 motion of the united masses after impact. Being equally distributed among 

 the 14 component parts of which these united masses consist, each part 

 will have a fourteenth of the whole motion. Hence, 196 being divided 

 by 14, we obtain the quotient 14, which is the velocity with which the whole 

 moves. 



In general, therefore, when two masses, moving in the same direction, im- 

 pinge one upon the other, and, after impact, move together, their common ve- 

 locity may be determined by the following rule : " Express the masses and 

 velocities by numbers in the usual way, and multiply the numbers expressing 

 the masses by the numbers which express the velocities ; the two products 

 thus obtained being added together, and their sum divided by the sum of the 

 numbers expressing the masses, the quotient will be the number expressing the 

 required velocity." 



From the preceding details, it appears that motion is not adequately estimated 

 by speed or velocity. For example, a certain mass, A, moving at a determinate 

 rate, has a certain quantity of motion. If another equal mass, B, be added to 

 A, and a similar velocity be given to it, as much more motion will evidently be 

 called into existence. In other words, the two equal masses A and B united 

 have twice as much motion as the single mass A had when moving alone, and 

 with the same speed. The same reasoning will show that three equal masses 

 will, with the same speed, have three times the motion of any one of them. In 

 general, therefore, the velocity being the same, the quantity of motion will al- 

 ways be increased or diminished in the same proportion as the mass moved is 

 increased or diminished. 



On the other hand, the quantity of motion does not depend on the mass only, 

 but also on the speed. If a certain determinate mass move with a certain 

 determinate speed, another equal mass which moves with twice the speed, 

 that is, which moves over twice the space in the same time, will have 

 twice the quantity of motion. In this manner, the mass being the same, 

 the quantity of motion will increase or diminish in the same proportion as the 

 velocity. 



The true estimate, then, of the quantity of motion is found by multiplying 

 together the numbers which express the mass and the velocity. Thus, in the 

 example which has been last given of the impact of masses, the quantities of 

 motion before and after impact appear to be as follow : 



