(66.) When two masses moving in opposite directions impinge and move together, their common velocity after impact may be found by the following rule:--“Multiply the numbers expressing the masses by those which express the velocities respectively, and subtract the lesser product from the greater; divide the remainder by the sum of the numbers expressing the masses, and the quotient will be the common velocity; the direction will be that of the mass which has the greater quantity of motion.” It may be shown without difficulty, that the example which we have just given obeys the law of “action and reaction.” Before impact. | After impact. | Mass of A 8 | Mass of A 8 Velocity of A 9 | Common velocity 3 ------------+ ----------- Quantity of motion } 8 × 9 or 72 | Quantity of motion } 8 × 3 or 24 in direction A C } | in direction A C } ------------+ ----------- Mass of B 6 | Mass of B 6 Velocity of B 5 | Common velocity 3 ------------+ ----------- Quantity of motion } 6 × 5 or 30 | Quantity of motion } 6 × 3 or 18 in direction B C } | in direction A C } ------------+ ----------- Hence it appears that the quantity of motion in the direction A C of which A has been deprived by the impact is 48, the difference between 72 and 24. On the other hand, B loses by the impact the quantity 30 in the direction B C, which is equivalent to receiving 30 in the direction A C. But it also acquires a quantity 18 in the direction A C, which, added to the former 30, gives a total of 48 received by B in the direction A C. Thus the same quantity of motion which A loses in the direction A C, is received by B in the same direction. The law of “action and reaction” is, therefore, fulfilled. This result may in like manner be generalised. Retaining the former symbols, the moving forces of A and B before impact will be A × _a_ and B × _b_ and their forces after impact will be A × _x_ and B × _x_. The force lost by A will therefore be A × _a_ - A × _x_. The mass B will have lost all the force B × _b_ which it had in its former direction, and will have received the force B × _x_ in the opposite direction. Therefore the actual force imparted to B by the collision will be B × _b_ + B × _x_. But since the force lost by A must be equal to that imparted to B, we shall have A × _a_ - A × _x_ = B × _b_ + B × _x_ and therefore (A + B) × _x_ = A × _a_ - B × _b_ and if the common velocity after impact be required, we have _x_ = (A × _a_ - B × _b_)/(A + B)