Fig. 133.
I will illustrate this in another way. Let a and b, Fig. 133, be two balls of clay of equal size hanging over a graduated arc. Now if b be let fall from the top of the arc, 6, on striking against a it gives half of its motion to a, and they both move on together. But how far will they go? To 3, on the other side of the arc. Why? Let the quantity of matter in each ball be called 1, and the motion of b 6. The momentum will therefore be 6. Now the momentum of the two together will be the same after the blow as that of b was before it. But the quantity of matter is twice as great, and must be called 2. Therefore the motion must be represented as 3, to make the momentum 6 (2×3). But suppose that b is twice as large as a. Falling from 6, its momentum would be represented by 12 (2×6). After it has struck a, the momentum of the two together would be the same as that of b before the stroke; but the quantity being 3, the motion would be represented by 4. They would therefore move to 4 on the arc.
202. Examples.—A few examples illustrating momentum as being compounded of quantity of matter and velocity will suffice. If a musket-ball of an ounce weight were so much spent as to move with only a velocity of a foot in a second, its force would be so small that if it hit any one it would do no harm. But a cannon-ball weighing a thousand ounces moving at this slow rate would have a very great force—equal, in fact, to the momentum of an ounce ball moving 1000 feet in a second.—If a plank push a man's foot against a wharf he will scarcely feel it; but if the plank, instead of being alone, is one of a thousand planks fastened together in a raft, and the whole move with the same velocity, the force will be increased a thousand-fold, and the plank will crush the foot. So, also, if the one plank when alone should move a thousand times as fast as the whole raft, the same result would follow.—So soft a substance as a candle can be fired through a board from the momentum given to it by an immense velocity.—Perhaps there is no better example of the great force given to a substance by an enormous velocity than we have in the wind. So light a thing is air that people think of it as almost nothing. But let it be set in rapid motion, and the velocity gives to it a force, a momentum, which will drive ships upon the shore, throw over buildings, and tear up trees by the roots. In this last example we see beautifully illustrated the meaning of the expression quantity of motion. In the moving air each particle does its share of the work in the destructive effects mentioned. Each particle, therefore, may be considered as a reservoir of motion, and the quantity of motion in any case depends upon the quantity which each particle has and the number of the particles.
203. Production of Great Velocities.—When there are no obstacles to motion great velocities may be produced by a single impulse. Thus at the beginning the Creator gave a single impulse to each of the heavenly bodies, producing enormous velocities, which continue unaltered year after year and age after age, because these bodies fly in their orbits through space where there is no resistance of any thing like air to retard the motion. But in all the motions that we see around us there are obstacles continually retarding them; and therefore no very rapid motion is produced by any single impulse, but a succession of impulses is required to accumulate sufficient momentum so to overcome the obstacles as to secure a great velocity. I will give a few examples in illustration. One of the best examples we have in the fall of bodies to the earth. You know that the greater the elevation from which a body falls the greater is its velocity, and therefore the greater the force with which it strikes. Why is this? If it fell because of a single impulse making it go toward the earth, this would not be the case, and if there were no air in the way the velocity would be uniform; but the resistance of the air would retard the velocity, so that if a number of bodies should receive the same impulse at different elevations, the one the farthest off would be the most retarded, and therefore come down slower than all the rest. In this case, the higher the elevation from which a man should fall the less would be the injury. But a body does not come to the ground by a single impulse, but by a succession of impulses, or rather a continued impulse. Every moment that the body is coming down it is drawn upon by the attraction of the earth, and this continued action of the cause of the motion makes it continually increase in rapidity. It is on the same principle of continued action that a man lifts his hammer high when he wishes to inflict a heavy blow. In this case both gravitation and the muscular power of the arm exert their force on the hammer through the whole space. A horse in kicking does the same thing, and by the great length of the leg the velocity given to the foot by this continued action of the muscles is very great. An arrow is not shot by a single momentary impulse of the bow-string, but the string, by following it through a considerable space, gives it a continued impulse. The action of gunpowder upon a bullet issuing from a gun is apparently an instantaneous and single impulse, but it is not really so. The great velocity given to the bullet is given to it by the continued impulse of the expansive force of gases produced from the powder, and it therefore depends much on the length of the barrel. If this be short, the force of the powder is not confined long enough to the bullet to give it a great velocity.
204. Arrest of Great Velocities.—As a continued force is required to produce great velocities, so a continued resistance is necessary to arrest them. It is by the gradual or continued resistance of the air that the motion of a cannon-ball is destroyed. Now if instead of this gradual resistance any hard substance, as a block of granite, were opposed to the progress of the ball, it would be at once broken asunder. We see then the reason that a hard substance of moderate thickness does not offer so effectual a resistance to a body moving very rapidly as some substance of a more yielding kind and of greater bulk. For example, a bale of cotton will arrest a ball which would pass through a plank, for the cotton yielding easily permits the force of the ball to be felt and resisted by a larger bulk, while the wood, not yielding, opposes but a small portion of its whole bulk to the force of the ball, and therefore does not arrest it; in other words, the momentum of the ball is communicated to a much larger quantity of matter in the cotton than in the wood. These principles afford a ready explanation of a feat which is sometimes performed. A man lies upon his back, and, having an anvil carefully placed upon his chest, allows some one to strike a heavy blow with a hammer upon the anvil, and no injury is received. Why? Because the momentum or force of the hammer is diffused throughout the bulk of the anvil, and then again through the bulk of the yielding chest. The man takes good care to have his lungs well filled with air at the moment of the blow, for this increases the bulk and elasticity of the chest, and thus promotes the diffusion of the momentum. If the blow of the hammer were received directly upon the chest great injury would be done, for the force would now be spent upon one small spot alone.—The principles above elucidated are applied by men instinctively in their common labors and efforts. You see a man catching bricks that are tossed to him. As he receives the bricks into his hands he lets his hands and the bricks move together a little way, so that he may gradually arrest the motion of the bricks. To do it suddenly would give him a painful lesson on momentum. So when a man jumps from a height he does not come to the ground in a straight position. This would cause a sudden and therefore a painful arrest of the motion of the whole body. To avoid this he comes to his feet with all the great joints of his body bent, so that the different portions approach the ground successively, his head having its motion arrested last.
Fig. 134.
205. Communication of Motion in Elastic Bodies.—Momentum is transferred from one body to another very differently in elastic from what it is in non-elastic bodies. As you saw in § 201, when one non-elastic body strikes upon another the momentum is divided between them, and both move on together. Now if a and b, Fig. 133, were elastic bodies, as ivory balls, and b should be let fall against a, it would give all its momentum to a. Therefore b would stop, and a would move on to the same height from which b came. The reason is, that the velocity lost by b and received by a is just double what it would be if the balls were non-elastic. For the same reason, if a and b, being elastic, meet each other from equal heights on the arc, they will both rebound, and return to the same heights from which they came. But if non-elastic they simply destroy each other's momentum and stop. The effect produced in the former case is just twice as great as in the latter, as you may see by reckoning on the arc. For the same reason, too, if you have a row of elastic balls, as in Fig. 134, and let a fall from the point i upon b, it will stop there; and communicating all its momentum to b, this momentum will pass from b to c, and so on through all the row of balls to e, the last one, which will fly off to the point h, at the same height with i, the point from which a fell. If b be held still, and a be let fall upon it, a will rebound to the height from which it fell, for then the compressed elastic spring (§ 39) of each ball, as b is immovable, communicates all the motion to a. It is for this reason that an elastic ball, on being thrown against any thing fixed, rebounds. If what it is thrown against be perfectly elastic it rebounds with a force equal to that with which it is thrown.
