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(65.) The bodies impinging were, in the last case, supposed to move in the same direction. We shall now consider the case in which they move in opposite directions.

First, let the masses A and B be supposed to be equal, and moving in opposite directions, with the same velocity. Let C, fig.5., be the point at which they meet. The equal motions in opposite directions will, in this case, destroy each other, and both masses will be reduced to a state of rest. Thus, the mass A loses all its motion in the direction AC, which it may be supposed to transfer to B at the moment of impact. But B having previously had an equal quantity of motion in the direction BC, will now have two equal motions impressed upon it, in directions immediately opposite; and these motions neutralising each other, the mass becomes quiescent. In this case, therefore, as in all the former examples, each body transfers to the other all the motion which it loses, consistently with the principle of “action and reaction.”

The masses A and B being still supposed equal, let them move towards C with different velocities. Let A move with the velocity 10, and B with the velocity 6. Of the 10 parts of motion with which A is endued, 6 being transferred to B, will destroy the equal velocity 6, which B has in the direction BC. The bodies will then move together in the direction CB, the four remaining parts of A’s motion being equally distributed between them. Each body will, therefore, have two parts of A’s original motion, and 2 therefore will be their common velocity after impact. In this case, A loses 8 of the 10 parts of its motion in the direction AC. On the other hand, B loses the entire of its 6 parts of motion in the direction BC, and receives 2 parts in the direction AC. This is equivalent to receiving 8 parts of A’s motion in the direction AC. Thus, according to the law of “action and reaction,” B receives exactly what A loses.