287. Let us apply the principle just stated to find the space through which the body will fall in the fourth second of its descent. The velocity at the beginning of the fourth second, that is at the end of the third second, is 3x32, that is 96. The velocity at the end of the fourth second is 4 × 32, that is 128. The half sum of 96 and 128 is 112, so that a body moving uniformly with the average velocity would describe 112 feet in a second. This, as we saw in Art. 91, is exactly the space through which a body falls in the fourth second of its descent, as it should be according to our principle.

288. Of the two numbers which thus present themselves in the laws of falling bodies, namely 16 and 32, we might take either as the representative of the force of gravity; but it is found most convenient to take 32 which denotes the velocity gained in the first second by a body falling freely. This number is very important in Mechanics; it is usually denoted by the letter g in books which discuss the mathematical theory of the subject. The strength of any other constant force, may be compared with that of gravity, by observing the appropriate number which now takes the place of 32. Thus at the surface of the sun for a falling body the number would be 27 times 32; the attraction of the sun at its surface being about 27 times the attraction of the earth at its surface. At the surface of the moon for a falling body the number would

1

be about of 32, that is rather more than 5.

6

289. We have already mentioned a contrivance, called Atwood's machine, by which we can exhibit a motion of the same kind as that of a body falling freely, but much slower, and so better adapted for observation: see Art. 140. Another case of such motion is that furnished by a body sliding down a smooth inclined plane. We have seen in Art. 246 that when a body is placed on an Inclined Plane it may be supported by a force acting along the Plane less than the weight of the body, namely by a Power having the same proportion to the Weight of the body as the height of the Plane bears to its length. This leads to the conclusion that a body will slide down the inclined

plane in the same manner as a body falls freely, but at a slower rate. Instead of the number 32 we must now take a smaller number, namely a number in the same proportion to 32 as the height of the plane is to its length.

1

8

For example, if the height of the plane is of its length the standard number with which we shall be concerned will be of 32, that is 4. A body sliding down such a plane would gain in the first second a velocity of 4 feet per second, and an equal additional velocity in every other second; and it would slide down through 2 feet in the first second. An important fact connected with this case of motion is that the velocity gained by a body in sliding down the inclined plane is precisely the same as would be gained by the body if it fell freely through the height of the plane.

290. Various interesting results are obtained by theory and may be verified by experiment respecting the motion of bodies down smooth inclined planes.

Thus, for example, let A be the highest point of a circle in a vertical plane, AB a diameter, AC any chord. Then the time of sliding down AC is equal to the time of falling freely down AB; so also the time of sliding down CB is equal to the same time.

A

B

291. Another example of motion of the same kind as that of a falling body is furnished by placing one body on a smooth horizontal table and allowing it to be drawn along the table by another body which descends vertically, the two bodies being connected by a string which passes over a pully at the edge of the table. Suppose for instance that the weight of the body on the table is 5 pounds, and the weight of the descending body 3 pounds. Then the mass to be moved is the sum of the two masses, and the corresponding weight is 8 pounds. But the weight of the body on the table is resisted by the table, and so it does

not produce any motion; and thus the weight of 3 pounds has to move all the mass instead of just moving itself.

3

Therefore the effect produced is of what would be pro

8

duced if the descending body were free; and the motion is like that of a falling body, only instead of the standard

number 32 we must use

3

8

of 32, that is 12.

XX. PROJECTILES.

292. In Art. 124 we have considered the motion of a body projected vertically upwards, and have shewn that the body will reach the height from which it would have had to fall in order to gain the velocity with which it was projected upwards. A few words may be given to the case of a body projected vertically downwards. A person might stand on a high tower and send a body vertically downwards, starting it say with a velocity of 64 feet per second. In this case the body starts with the velocity which would be gained in falling for two seconds, and the subsequent motion is precisely the same as that of a body which falls freely, but which began its descent just two seconds before we turned our attention to it. As in Art. 126 we must notice that during the motion, that is after the body has been projected, the only force acting is the force of gravity.

293. We have hitherto considered only motion in a straight line, but daily observation presents us with examples of other kinds of motion. The most familiar case is that in which a body is started in some direction neither vertically upwards nor vertically downwards, and is left to move under the action of gravity. As examples we may take a cricket-ball thrown by the hand, an arrow shot from a bow, and a ball shot from a cannon. A body thus projected and left to the action of gravity is called a Proectile

A

294. Let a body be projected from the point A, in any direction which is not vertical; let AT be the space which would be described by the body in any assigned time if gravity did not act, so that AT is the direction of projection. Draw AM vertically downwards, equal to the space through which a body would fall from rest in the assigned time under the action of M gravity. Complete the parallelogram ATPM; then P, the corner opposite to A, will be the place N of the body at the end of the as

T

signed time. For by the Second Law of Motion gravity will communicate the same vertical velocity to the body as it would if the body had not received any other velocity. Thus at any instant there will be the same vertical velocity as if there had been no velocity parallel to AT, and the same velocity parallel to AT as if there had been no vertical velocity. Therefore the spaces described parallel to AT and AM respectively will be the same as if each alone had been described. Thus P will be the place of the body at the end of the assigned time.

295. It is obvious that by the method just given we may determine the position which the projectile has at any assigned instant; and if we go through the process for a great number of different instants, marking on a piece of paper the places obtained, we shall obtain a good representation of the path of the projectile. It is found to be a curve which mathematicians call a parabola, and of which they have discovered many interesting properties. But we do not assume that the reader has at present studied the nature of this curve. Some idea of the form of the curve may be gained by watching the flight of an arrow. Or suppose we make a small hole in the lower part of the side of a barrel full of water; the drops of water are forced out and become projectiles, and as one follows another we have a continued stream which takes the form of a parabola.

296. The parabola is not a closed curve like a circle, but stretches on without end; and in this respect it resembles a straight line. Suppose one arrow to be shot nearly vertically upwards, and another to be shot very obliquely; then at first sight the two paths may seem to be not much alike. The reason for the apparent diversity is that the two paths are not of corresponding extent; but in reality all parabolas are similar, that is, a portion of one parabola is an exact copy of the corresponding portion of any other, though it may be on a larger or a smaller scale.

297. Return to the diagram of Art. 294. Produce AM to N, so that MN is equal to AM, and complete the parallelogram ATQN; then when the body is at P it will be moving for an instant in the direction parallel to the diagonal AQ Suppose, for example, that 3 seconds elapse, after starting, before the body is at P; then AT represents the space through which the body would move in 3 seconds if gravity did not act: thus if the original velocity is 40 feet per second AT represents 3 x 40 feet. Also AM represents the space fallen through under the action of gravity in 3 seconds, so that AM=9 × 16; and therefore AN=9 × 32. Hence AT bears the same proportion to AN as 3×40 bears to 9 x 32, that is as 40 bears to 3 x 32. But 40 represents the original velocity along AT, and 3 × 32 represents the vertical velocity given by gravity in 3 seconds. Hence AT and AN are proportional to the component velocities of the body at the end of 3 seconds, and are in the directions of these velocities respectively. Hence when the body is at P the direction of the resultant velocity is parallel to AQ; and the magnitude of the resultant velocity bears the same proportion to that of the original velocity as AQ bears to AT.

298. While the projectile is in motion the only force acting on it is that of gravity; if at the instant the projectile is at P this force were to cease acting, the body would thenceforward move on uniformly in the direction, and with the velocity, just determined.

299. Another method of treating the problem of projectiles may be briefly noticed, as it is found very advantageous in mathematical calculations. This consists in