FIG. 23, deck, and that the ball would therefore alight at some distance behind the mast. But it does not. Let AB represent the position of the mast when the ball is set free, and ab the position it has arrived at when the ball strikes the deck. The ball, it is to be observed, has two motions; the one, a progressive motion in common with the ship, which alone would have carried it, in the time occupied by its descent, from A to a; the other, a motion produced by gravity, which alone would have carried it, in the same time, from A to B. Its real course is therefore from A to b; not, however, in a straight line, but in a curve, because the motion in falling is not uniform, as will be explained. when we come to speak of gravity. From these examples, and many others of a like nature, it appears that when more than one force acts on a body, each produces an effect, in its own direction, equal to what it would produce if it acted alone, and the real motion of the body is the aggregate of these effects. III. Every one knows that a moving body strikes with a greater or less force any obstacle that may lie in its way. But most people overlook the fact that the body which is struck always returns an equal blow. If a boy, by an incautious motion, knocks his head against his neighbour's, the two will suffer alike, unless one be struck on a more tender part than the other. When a boy flies a kite, the string pulls equally both ways, the kite and the boy's hand communicating to it exactly the same force. A bird, in flying, strikes the air with its wings, and the air, by its return blow, gives the body of the bird an upward and forward impulse. The fins of a fish, the arms of a swimmer, and the oars of a boat receive a similar impulse from the water. In all these instances, a certain force, acting in one direction, is accompanied by a corresponding force reacting in the opposite direction. Now the third law of motion asserts that this must always be the case. It is this: Action and reaction must be equal and contrary; or, the forces exerted A B by two bodies upon one another must be equal, and their direc tions must be opposite. Such are the three great principles which the immortal Newton, guided by the partially successful gropings of previous philosophers, was the first to announce with the prominence they deserve. Simple as they are, their establishment has led to some of the grandest discoveries recorded in the history of science. MOTION PRODUCED OR MODIFIED BY GRAVITY. THE whole attraction exerted by the different parts of the earth's mass, may be considered as directed towards its centre; for, from the symmetry of its shape, the forces which draw a body towards opposite sides of the centre must obviously balance each other. A falling body, then, moves directly towards the centre of the earth. A ball, suspended by a string, has a tendency to move towards the same point, and accordingly will pull the string in that direction. A mason's plumb-line is a good example; it hangs in what we call a vertical line, a line which, if produced, would pass through the earth's centre. There all vertical lines would ultimately meet, so that they are not quite parallel; but any two of them, moderately near each other at the earth's surface, may for practical purposes be regarded as parallel lines. It is quite in accordance with the testimony of our senses that falling bodies move in parallel or very nearly parallel directions, but we are not so ready to believe that all bodies, heavy and light, fall with equal rapidity. Nor do they in point of fact; yet they would do so if acted upon by gravity alone, and not retarded in their fall by the resistance of the air. There is a machine called an air-pump, by which nearly the whole of the air can be exhausted from a vessel of any size or shape. If a long glass tube be thus emptied of the air it contains, and then closed so as to be air-tight, it will be seen that a guinea and a feather, or any similar bodies previously placed in it, will fall from one end to the other exactly in the same time. This has long been a rather celebrated experiment, but there is a simpler one which any boy may try for himself. He has only to get a penny, and cut a piece of paper into the same shape, but a little smaller. If he drop both from his hand at once, the penny will strike the ground long before the paper. But let him place the paper on the upper side of the penny, taking care that it lie flat, and that its edge do not overlap the penny at any point. If they be dropped in this posi tion, so that the lighter body shall not have to overcome the resistance of the air, it will be found that both will reach the ground at the same time. It is clear that the velocity of a falling body cannot be uniform. A stone dropped from the top of a high tower, is acted on by gravity during the whole time of its descent, and its speed is therefore constantly increasing.* Every successive instant it receives a new impulse, while still retaining the force it has already acquired. Thus, at the end of the first second after it has begun to fall, it is found to have attained a velocity of rather more than 32 feet per second. In course of the next second, it acquires as much more, so that at the end of two seconds its velocity is doubled. At the end of three, it will be tripled; and so The velocity of a falling body is therefore proportional to the time since it began to fall. The space, however, through which it falls, increases much more rapidly. It would be proportional to the time, if the velocity were uniform; but since the velocity itself increases in the same proportion as the time, the space must increase in the same proportion as the square of the time. This inference is verified by many ingenious experiments, which cannot here be described. It is found that a body, falling freely from a state of rest, and not resisted by the air, would describe a space of 16 feet 1 inch in one second; a space four times as large in two seconds; a space nine times as large in on. * It is for this reason that gravity is called an accelerating force, and the motion it produces an accelerated motion. three seconds; and so on. Hence the velocity of a falling body, and the force with which it strikes the ground, are not proportional, as is often supposed, to the height from which it has fallen, but only to the time which its fall has occupied. If, instead of being merely allowed to fall, a body is projected vertically downwards, with a certain force, that force, acting along with, and aiding the force of gravity, will bring the body sooner, and with greater velocity, to the ground. The velocity produced by the joint action of the two forces, is the sum of the velocities which they would produce if they acted separately. Let us suppose, on the other hand, that the body is projected vertically upwards. In that case, the force of projection will be opposed by gravity. The velocity of the body, at any instant during its ascent, will be the difference between the upward velocity with which it was projected, and the downward velocity which gravity would have imparted to it if it had from the first been free to fall. This downward velocity increases as the time increases, until at last it becomes exactly equal to the velocity of projection. The body then ceases to rise, and being now free, will fall again in obedience to the force of gravity alone. It will be seen, on a little consideration, that its ascent and descent must occupy exactly equal times, and that every point in its course will be passed twice, each time with the same velocity. All this is on the supposition that the body is not resisted by the air, which, however, in reality, will oppose its progress both in ascending and descending. There is yet another case to be noticed, that of a body projected in a direction not vertical. Suppose a body A (fig. 24) to be projected in the direction A B with such a velocity as would carry it to the point B in five seconds, if gravity did not act upon it. But in five seconds the same body would fall, if acted upon by gravity alone, a little more than 400 feet. Let AC represent that distance. At the end of the five seconds, then, the body will be found, in accordance with a principle formerly explained,* at D, the opposite angle of the * Page 246. FIG. 24. parallelogram of which A B and A C are adjacent sides. It does not, however, pass along the diagonal A D, for the effect of gravity is at first comparatively small, and draws it but little out of the straight line A B, in which it tends to move at the moment of projection. But since the effect of gravity increases as the square of the time, it gradually diverges from that line more and more rapidly, and so describes the curve AEFD, until some obstacle interrupts its progress. If the ground is level, as A F, the body will be stopped by it at the point F. A body moving in this way is called a projectile, and the curve it describes a parabola. A stone thrown slantingly upwards, with moderate force, very well illustrates this kind of motion. A rifle-ball is also a projectile; but, from its great velocity, it meets with so strong a resistance from the air, that it deviates very much from the path it would otherwise pursue. There is, however, no difficulty in seeing why a rifle is not directed exactly to the point which the marksman intends to hit, but considerably above it, allowance being made for the effect of gravity upon the ball as it speeds on to its destination. A E B CENTRE OF GRAVITY. THE force of gravity acting on a body causes it to fall; or, if it is supported so that it cannot fall, it will exert a cer tain pressure on the body that supports it. But we may observe that a body which is not free to fall right downwards, often tumbles to one side, and so assumes a new position. If, for example, we wish to lay down an egg on a level table, we can put it in a great many positions in which it will not remain at rest. A cart on a level road |