Rule for finding the number of seconds occupied in the fall: Divide the number of feet fallen through by 16 and extract the square root of the result. Then if we multiply the result thus obtained by 32 we obtain the velocity at the end of the time. Or we may obtain this velocity by the following Rule which will be found on trial to agree with the former: Multiply the number of feet fallen through by 64, and extract the square root of the product. The last Rule is important and often wanted in practice.

VII. MASS AND MOMENTUM.

128. Suppose we take two bodies of the same size and shape, say a cricket ball and an iron ball just as big; we find that the iron ball presses more strongly than the cricket ball on the hand which holds them: in fact the iron ball weighs more than the cricket ball, Now we use the word matter to express the substance, material, or stuff of which bodies are composed; and we use the word mass as an abbreviation for quantity of matter. We also take it for granted that at the same place on the earth's surface the mass of bodies is proportional to their weight.

129. The reader will naturally be led to think that as mass is proportional to weight it is unnecessary to introduce the word mass. But as we proceed it will be found very convenient to have this word expressing something which belongs to the body, and remains unchanged when the body is taken from one place to another. We have said that at the same place on the earth's surface the mass is proportional to the weight, and it is important to bear in mind this condition, for the weight of a body is not the same at all places. If we use a pair of scales to weigh a body in the ordinary manner, we shall find no difference in the number of pounds and ounces which we call the weight of the body when we take the scales to various places. foolish person being told that bodies weighed less when taken to a height above the earth's surface than they did at the surface, declared that the statement was untrue, for he had weighed a body most carefully in the cellar and in the attic of his house, and found no difference in the two cases. He had misunderstood what he had been told, and

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which we may explain as follows. Suppose a string just strong enough to bear at London without breaking a certain piece of lead fastened to the end of it; then at the equator it would bear a rather larger piece without breaking, while at the pole it would not bear quite so much. If a string could bear at the pole the weight of 200 coins all exactly alike, then at the equator it would bear the weight of about 201 of them; or in other words the weight of any body is diminished by about in passing from the pole to the equator. The diminution is another consequence of the same cause as that which operates in Art. 98; where the result is a diminution of about 1 inch in 16 feet with respect to the fall of a heavy body in a second. Instead of weighing a body in scales we may make use of one of the contrivances by which the result is ascertained by noticing how far the body will bend a spring; then it will be found, if we employ a very delicate spring, that the weight is less in places which are nearer to the equator than in those which are further from it.

130. In examining questions about motion we soon learn that we have to pay attention to two things, the mass in motion, and the velocity with which it is moving. Thus the mischief and destruction which a cannon ball produces increase both as the mass of the cannon ball increases and as the velocity with which it moves increases; and a similar remark holds with respect to the disaster of a collision between ships, or between railway trains. An iceberg, though moving with very small velocity, may produce a great effect by its vast mass. Accordingly we are led to the important idea which we express by the word momentum; this means the product of the mass into the velocity. Thus if one body has a mass which we denote by 2, and a velocity which we denote by 3, the momentum is 2 x 3, that is 6; hence another body which has a mass 3 and a velocity 2 will have the same momentum; and a third body which has a mass 4 and velocity 6 will have a momentum 24, which is four times as great as in the former cases. The word momentum is one of those which unscientific people employ in various senses, so that the reader must bear in mind the strict meaning which we give to it.

131. We will now repeat the second Law of Motion. Change of motion is proportional to the acting force, and takes place in the direction of the straight line in which the force acts. By motion here we are to understand motion as measured by momentum; and with this explanation we need not restrict ourselves to the case of one body and one force, but may if we please take more complex cases in which different bodies and different forces

occur.

132. The effect of force then is to give velocity to bodies, and we measure the effect by the momentum produced. Hence if we have a certain force at our disposal we can produce only a certain amount of momentum; if we operate on a heavier body we produce a less velocity than if we operate on a lighter body. Thus if a blow will give a certain velocity to a ball, the same blow applied to a ball of double weight will give half the former velocity. Now it will be seen that the force of gravity differs remarkably in one respect from the forces of men, of animals, of wind, of water, and of steam with which we are familiar. In all the latter cases we are accustomed to see a less velocity produced according as the body in which it is produced is greater. But when bodies fall to the ground, whether they are large or small they acquire equal velocities in falling for the same time. The fact is that the force of gravity is not of a fixed amount for all bodies, but varies in proportion to the mass moved. If a double mass has to be moved the force of gravity puts forth as it were a double energy; or in other words the force of gravity acts on each of the two equal halves of the double mass just as if the other half did not exist.

133. We can now give some explanation of the fact noticed in Art. 99, that the resistance of the air interferes more with the motion of light bodies than with the motion of heavy bodies. Let us suppose a hollow ball made of very thin iron, and a solid ball of the same size also made of iron. As we have just remarked, the force of gravity will give the same velocity to one ball as to the other in the same time, so that, setting aside the resistance of the air, the two balls would fall through equal spaces in the

same time. Now consider the resistance of the air; this cannot depend in any way on the nature of the inside of the balls, and so must be the same on two balls of the same size, shape, and texture of surface, if they move with the same velocity. But by Art. 132 this force would produce less velocity in the solid ball than in the hollow ball; and so in the actual case we may readily suppose that it will take away much less of the downward velocity of the solid ball than of the hollow ball.

134. An example will illustrate the difference of the influence of the resistance of the air on bodies_differing only in size. Suppose, for example, two cannon balls, one 4 inches in diameter and the other 5, but formed of the same material. The masses of the balls are in the same proportion as the cubes of the diameters, that is in the proportion of 64 to 125. It appears by experiment and theory that the resistances of the air are in the same proportion as the squares of the diameters, that is in the proportion of 16 to 25, that is in the proportion of 80 to 125. Hence we see that the resistance on the smaller ball bears to that on the larger ball a greater proportion than the mass of the smaller ball bears to that of the larger; and so the resistance exercises more influence on the smaller than on the larger ball, supposing the velocities equal.

135. We have not given a very full account of the influence of the resistance of the air because we have not attended to the way in which the resistance depends on the velocity of the moving body. In reality the resistance increases very rapidly as the velocity of the moving body increases. For an example, it has been found that under certain circumstances the range of a cannon ball would be 23000 feet if there were no such resistance, while it was in fact about 6400 feet. Another example is furnished by an experiment with a railway engine. The engine was started down an inclined plane with a velocity of 45 miles an hour; the velocity gradually diminished until it became 32 miles an hour and remained at that. Thus the resistance of the air together with that caused by the want of perfect smoothness in the wheels and iron rails just balanced the influence of the force of gravity in urging the engine down the plane and maintained uniform velocity.

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VIII. THIRD LAW OF MOTION.

136. Third Law of Motion. To every action there is always an equal and contrary reaction: or the mutual actions of any two bodies are always equal and oppositely directed in the same straight line.

Newton gives three illustrations of this Law:

If any one presses a stone with his finger, his finger is also pressed by the stone.

If a horse draws a stone fastened to a rope, the horse is drawn backwards, so to speak, equally towards the stone. If one body impinges on another body and changes the motion of the other body, its own motion experiences an equal change in the opposite direction.

In the third illustration motion is to be measured by momentum as in all cases. We shall return to the discus

sion of this illustration hereafter.

137. One of the most important examples of this Law is furnished by the attraction of bodies. The earth, for instance, attracts a body, and that body attracts the earth again with equal power. Thus when the earth produces velocity in a falling body that falling body also produces velocity in the earth, although the latter velocity is so small as to be imperceptible. For, according to the third Law of Motion, the stone gives to the earth as much momentum as the earth gives to the stone, and as the mass of the earth is incomparably greater than that of the stone the velocity given to the earth is incomparably less than that given to the stone. In the science of Astronomy the mutual attraction of bodies is a principle of supreme importance; the earth, for instance, attracts the moon, and the moon attracts the earth again with equal power.

138. The fact that not merely the earth as a whole attracts, but that distinct portions of the earth also do so, has been made obvious by noticing the action of moun