MECHANICS. 



attracts the mass of the earth towards 

 it, but attracts it with exactly as much 

 force as that by which the earth at- 

 tracts the descending body. Why then, 

 it will be asked, is not 'the rapid ap- 

 proach of the earth to meet the de- 

 scending body perceptible ? To explain 

 this we must go into some details. 



(16.) If two bodies A and B be mov- 

 ing with the same velocity, the forces 

 with which they move will be equal 

 provided their masses or quantities of 

 matter be equal, but not otherwise. If 

 the mass of A be greater than the mass 

 of B its force will be greater in the same 

 proportion. This will be very evident 

 if we consider the forces with which 

 they would strike any obstacle opposed 

 to them. If B be a musket-ball and A 

 be a cannon-ball of one hundred times 

 the weight, both being projected with 

 the same speed, A will strike any ob- 

 stacle with one hundred times the force 

 with which B would strike it. In ge- 

 neral, then, " when the velocities with 

 which bodies are moved are the same, 

 their forces are proportional to their 

 masses or quantities of matter. 1 ' 



Now, let us suppose that the masses 

 of the bodies A and B are equal, but 

 that they move with unequal velocities ; 

 that is, that they move through differ- 

 ent spaces in the same time. Let the 

 space described in one second by the 

 body A be a, and let the space described 

 in the same time by the body B be b ; 

 these spaces are called the velocities of 

 the bodies. The equal bodies thus mov- 

 ing with different velocities will move 

 with different forces. It is evident that 

 the body which has the greater velocity 

 will have the greater force; and that 

 also in the same proportion as its velo- 

 city is greater. If two equal bullets be 

 successively projected from the same 

 gun, but with different charges of pow- 

 der, that which is projected by the 

 stronger charge will strike the mark 

 with a proportionally greater force. But 

 in this case the only difference in the 

 motions of the bullets is, that one has 

 a greater velocity than the other. B 

 Hence we perceive that " when equal * 

 masses are in motion their forces are 

 proportional to their velocities." 



(17.) We have thus separately con- 

 sidered the cases in which unequal 

 masses are moved with equal velocities, 

 and in which equal masses are moved 

 unequal velocities; and we have seen 

 that the forces are, in the one case, 

 proportional to the masses, and in the 



other, to the velocities. Now, if unequal 

 masses be moved with unequal veloci- 

 ties, it is natural to expect that wa 

 should, in comparing the forces, take 

 into account both the velocities and the 

 masses. It appears that the moving 

 force of a body may be increased or 

 diminished by increasing or diminishing 

 either its mass, or its velocity, or both. 

 In fact, if the number representing the 

 mass be multiplied by the number repre- 

 senting the velocity, the product thus 

 obtained will represent the moving force. 

 Thus, if the masses of two bodies A and 

 B be in the ratio of the numbers 8 and 

 5, and the velocities of these bodies be 

 in the ratio of the numbers 7 and 3, 

 their moving forces are as the product 

 of 8 and 7 to the product ;of 5 and 3, 

 that is, as 56 to 15. It appears, there- 

 fore, that in this instance the force 

 of A bears a much higher ratio to the 

 force of B than either the mass of A 

 bears to the mass of B, or the velocity 

 of A to the velocity of B ; the reason of 

 which is, that the mass and velocity 

 conspire in imparting to A a supe- 

 rior moving force. In general, then, 

 we conclude, " that the moving forces of 

 bodies are proportional to the products 

 of their masses and their velocities." 



(18.) Since then the moving force of a 

 body depends conjointly on its mass and 

 its velocity, it necessarily follows that 

 if, while we increase its velocity in any 

 proportion, we diminish its mass in the 

 same proportion, its moving force will 

 be the same ; for it will lose as much 

 force by the diminution of its mass as it 

 gains by the increase of its velocity. In 

 like manner, if, while we increase its 

 mass, we diminish, in the same propor- 

 tion, its velocity, the moving force will 

 be unaltered ; for as much will be lost 

 by the diminished velocity as will be 

 gained by the increased mass. 



(19.) The several theorems which 

 we have just expressed, relative to the 

 forces of bodies in motion, do not rest 

 entirely upon reasoning, but may very 

 easily be submitted to the test of actual 

 experiment. 



Let two strings be attached to the cen- 

 tre C of a graduated arch X Y ; and let 

 two balls AB, formed of clay or any 

 other inelastic substance, be suspended 

 so as to hang in contact at the middle of 

 the arch. Let us first suppose that the 

 balls are equal: if they be separated, 

 moved from the middle of the arch in 

 opposite directions towards its extre- 

 mities, and permitted to descend from 



