right angles to CC'; also resolve the velocity of the ball whose centre is Cinto two components, one along CC, and the other at right angles to CC'. Then the velocities along CC are changed in precisely the same way as if the balls moving with these alone came into direct collision; and the velocities at right angles to CC are not affected at all; that is they remain the same for each ball after collision as before. Since we thus know the two component velocities of the ball whose centre is C, we can find the resultant velocity after collision, and the direction, CB, of this velocity. Similarly we can find the resultant velocity after collision of the other ball, and its direction C'B'.

It is often convenient to resolve velocities into components in the manner just exemplified; the method is the same as for resolving forces: see Art. 156.

284. An important case of oblique collision is that in which a moving ball strikes a fixed plane.

Let AC represent the direction in which the ball moves before it strikes the fixed plane at C; let CD be at right angles to the plane. After striking the plane the ball will go off in some direction which we denote by CB. The angle ACD is called the

A

B

angle of incidence, and the angle BCD the angle of reflection. If the ball and the fixed plane are perfectly elastic these angles are equal, and the velocity of the ball after collision is equal to the velocity before. If the ball and the fixed plane are imperfectly elastic the angle of reflec tion is greater than the angle of incidence, the relation between the two depending on the index of elasticity. In the case in which the ball and the fixed plane are inelastic the angle of reflection is a right angle, so that the ball after collision moves close to the plane. The velocity after collision is always less than the velocity before collision, except when the ball and the fixed plane are perfectly elastic.

285. Many remarkable results are obtained by the collision of balls on a billiard-table, which the principles we have stated would not be sufficient to explain. These results depend on two circumstances which we have not considered, namely the rotation of the balls, and the friction between the balls, and between the balls and the table: the theory of such results would be altogether beyond the present work.

XIX. MOTION DOWN AN INCLINED PLANE.

286. We have already spoken about the motion of a body falling freely, but we will now make a few additional remarks on the subject. The motion in this case is said to be uniformly accelerated: this means that in successive equal intervals of time the velocity of the falling body receives equal additions. The laws of the motion involve, as we saw, two numbers, namely 16 which expresses the number of feet fallen through in the first second of time, and 32 which expresses in feet per second the velocity at the end of the first second. The first number is half the second, and the reason for this may be seen without difficulty. The velocity increases in the same proportion as the time, and in the first second the velocity begins with the value 0 and ends with the value 32. Hence 16 may be called the average velocity; for instance at the end of the first tenth of a second the body is falling with the

1

velocity of 32, and at the end of nine-tenths of the

10

9

second it is falling with the velocity of 32: the sum of

10

these two velocities is 32, so that the half sum is 16. It is easy to admit that when the velocity increases or decreases uniformly as the time increases, then the space described in a given time is just the same as would be described by a body moving during that time uniformly with the average velocity, that is with a velocity equal to half the sum of the velocities at the beginning and the end of the given time.

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

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

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

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

3

number 32 we must use of 32, that is 12.

8

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 Projectile.