than that to which we here confine ourselves. The caution as to the kind of motion we are considering is expressed by some writers by speaking in the Laws of Motion of particles instead of bodies.

124. A few more remarks relating to falling bodies will be useful in further illustration of the second Law of Motion. Suppose a body to fall from a certain height to the ground; then if the body be sent straight upwards, starting with a velocity equal to that with which it reached the ground, it will just reach the height from which it fell, taking the same time in the ascent as it did in the fall. This may be easily seen from a particular instance. Suppose that the time of the fall is four seconds; then by Art. 92 the body reaches the ground with the velocity of 128 feet per second. Start the body straight upwards with this velocity; then if gravity did not act the body would move through 128 feet in one second, and still retain at the end of the second the velocity with which it started: this follows from the first Law of Motion. But during this second gravity acts, and in the direction just contrary to the motion; and in virtue of this a downward velocity of 32 feet per second is given to the body, and also the body is drawn down through 16 feet. In consequence of this the body really ascends upwards through 112 feet, and has at the end of the second an upward velocity of 96 feet per second. That is the body has ascended through just the space which a body would fall through during the fourth second of its motion, and it has a velocity upwards just equal to the downward velocity of a body at the end of the third second of its fall. In precisely the same manner we can shew that in the next second the body will ascend through 80 feet, and will have at the end of the second an upward velocity of 64 feet per second; that is it will ascend through just the space which a body would fall through during the third second of its fall, and it has a velocity upwards just equal to the downward velocity of a body at the end of the second second of its fall. Proceeding in this way we find that the body just reaches in four seconds the height from which it fell, and it has then no velocity, so that it goes no higher.

125. In the same way as the example of the preceding Article was treated we may treat any similar example. It will be observed that the following proposition becomes evident from the course of the discussion. Suppose a body to fall from a certain point to the ground and to be started upwards with a velocity equal to that with which it reached the ground, then the velocity on reaching to any height is equal to that of the falling body at the same height, though in the opposite direction. We may also shew that the following Rule will give us the height which the body will reach in any assigned time: Calculate the space through which a body would have moved uniformly in that time and from it subtract the space through which a body would have fallen from rest in that time; the remainder gives the height required. For instance, returning to the Example of Art. 124, find the height reached in two seconds. A body moving uniformly with the velocity of 128 feet per second, will in two seconds describe 128 × 2 feet, that is 256 feet. And in two seconds a body would fall through 16×2×2 feet, that is through 64 feet. Now 256-64=192; and 192=112+80, that is 192 feet is the height of the body at the end of two seconds by Art. 124. It must be remembered that in this and the preceding Article we neglect the influence of the resistance of the air.

126. While a body is falling the only force acting on it is gravity, and so also while a body is rising the only force acting is gravity, which acts in the direction contrary to motion. It is quite true that in the latter case some force must have acted just at the beginning of the ascent to start the body, but still during the ascent the only force acting is gravity. This may appear a simple remark, but it is necessary to draw attention to it, because some popular books are very erroneous as to the matter.

127. If we know how long a body has been falling we can immediately determine the space through which it has fallen, by Art. 88; and we can determine the velocity which it has at the end of that time by Art. 92. From the Rules given in these two Articles various others can be deduced by processes which do not require more than common Arithmetic. Thus from Art. 88 we may deduce the following

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. A 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

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 2×3, 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