50

ཋ་

A e

f

Composition of Velocities.

B

in the one direction, corresponding distances, e ƒ, g h, &c., in the other; so that its course will clearly be along the diago. nal or middle line of the parallelogram, A B C D, whose sides, A B and A D, represent the directions and rates of the simultaneous motions.

D

Fig. 3.

If the velocities in the two

crossing directions be equal, then the resulting motion must be exactly midway between these directions; for there is no reason why it should be nearer to the one than the other.

Thus a boat's motion is the resultant of equal velocities communicated by the oars on each side.

This explains also why a bird flying, or a man swimming, holds a perfectly straight course.

D

116. In ascending a staircase or the side of a hill we execute the

Fig. 4.

B

resultant of a combined vertical and horizontal motion. We have an unlimited command of horizontal, but only a very limited command of continued vertical movement. Yet, by combining the two, we produce a continued resultant motion, and so obtain a continued vertical one also.

This simple rule for finding the actual motion, or resultant, as it is termed, of two co-existing motions is known as the Parallelogram of motions or velocities. It is of the utmost importance, both in theory and practice.

117. On the same principle we find the conjoint effect of three or more velocities which a body may possess simultaneously.

A

A train may possess a north, an east, and a vertically upward motion at the same instant. If we take O B and O C to represent the north and east motions of the train, and O A to represent c the vertical one, or the rate at which it is rising perpendicularly; then, completing the solid figure, A B C D, by drawing parallels through A, B, and C, we shall have the resultant motion represented in amount and in direction by O D. In other words, if a body were urged simultaneously to move in

B

Fig. 5.

Resolution of Velocities.

51

each of the directions O A, O B, O C, at rates represented by those lines, then it would actually move along O D at a rate represented on the sanie scale by the length of O D.

"Resolution of motion or velocity."

118. We are but stating the same principle from the other side when we say that any motion or velocity may be regarded as equivalent to a compound motion taking place in two or more directions at once.

A meteor moving in a slanting direction from the north-east to the earth may be supposed to have three co-existing velocities—one from north to south, another from east to west, and a third vertically down.

When we consider the motion as thus broken up into three coexisting motions we are said to resolve it into its components in any given directions.

In cases of rectilinear motion the value of this mode of regarding it is not so apparent ; but when we have to deal with curved motion, as with that of a stone thrown obliquely, it becomes absolutely necessary to our calculation of the rate and direction of motion at any time that we consider the body as possessing at once two distinct velocities, each independent of the other.

119. The proportion of motion that goes to each direction is obtained very readily by the rule of the parellelogram of velocities. Thus, a train moving in the direction A C at the rate of six miles an hour is approaching

a point, P, in one direction at a certain rate, and at the same time approaching a point, Q, in another direction at a certain rate.

These rates may be found in this way :We take a line, A C, to represent the actual motion, and draw parallels through C to the

B

P

directions in which P

Fig. 6.

and Q lie from the starting point of the train. We thus form a representative parallelogram, A B C D ; so that if A C measure six inches, while A B is four, and A D three inches, the train will be approaching P at the rate of four, and Q at the rate of three miles an hour.

52

Curved Motion-Motion of Projectiles.

"Parabolic path of a projectile."

120. The resultant of two or more velocities is straight when the velocity in each direction keeps unchanged; but if the rates of motion do not remain the same comparatively, then the actual motion will be curved.

A ferry-boat rowed at a uniform rate across a river, and borne at a uniform rate down by its flow, will describe an intermediate rectilinear path.

On the other hand, a bullet shot obliquely from a rifle, say in the

L

F

B

Fig. 7.

direction repre

sented by A K,

possesses simultaneously, first, a sensibly uniform velocity, due to the explosion in the direction A K, and, second, a constantly increasing velocity

in the vertical direction, due to the pulling of the earth. The result is a curved motion of the bullet along A D F B. During the time it would have gone from A to C in virtue of the velocity imparted by the shot, it will have fallen towards the earth by an amount, CD; and during the time when it would have gone twice as far-from A to E-in virtue of this original motion, it will have fallen through a distance, E F, not twice, but four times as great as C D, and the downward velocity at F will be twice what it was at D. Thus, the route taken by the rifle-ball is not straight, but curved, and this is the explanation of the well-known curved course always taken by a stone thrown, an arrow shot, or a fountain playing obliquely.

A jet of water, or the fiery trail of a rocket, exhibits to the eye the parabola or curve described by a body thus projected.

121. It is, in every case, this parabolic course which a body will pursue if it be subject to a uniformly increasing velocity in one direction and a constant one in another. But, with a different relation between the component velocities in any two directions, a different curvature of actual path will obtain.

Force-Expressed and Potential.

53

122. We might keep the idea of matter in abeyance, and pursue this study of motion in the abstract much farther. It is of great importance in all branches of natural philosophy, as well as in those practical sciences-such as gunnery, mechanism, &c.-where motion is the most prominent feature; just as the abstract study of the rules of arithmetic is of the utmost consequence for the business and trades of practical life, where they have incessant application.

But for the nature of this work the further development of these abstract principles is unsuitable, and we accordingly proceed to consider motion as we find it expressed in nature, namely, rolled up in matter, the two together forming what may be called the factors of force.

"Force."

123. A rifle bullet in motion possesses force; when at rest, it is powerless.

If the hand be laid on the table, and a weight be placed on it, it presses with a certain force, which is simply the pull that the earth exerts on the weight. A magnet draws a piece of iron, or the earth draws a magnetic needle into a north and south direction with a certain force, which we can measure. Iron wire resists being pulled asunder with very great force.

In these and in all other instances, the idea conveyed by the word force is a tendency to put matter in motion, whether. the effect of that tendency be manifested or not.

The motion, as will be more particularly considered afterwards, may not be one of a large visible mass; it may be among the atoms or molecules of a body, and consequently so minute that it escapes our direct perception. We cannot, for instance, see the motion of the molecules of steam, yet we know that a multitude of such minute motions combine to move the piston-rod of the engine, and this in turn to move perhaps a score of waggons.

124. Moving matter, then, or matter in motion is the expression of force. Yet we may have force really existing unexpressed, possible, or potential, though ready to appear as motion at any

moment.

As, for example, a boulder on the face of a hill may be kept from moving merely by a small stone in front of it. It may lie for years and never exhibit its force, while the mere removal of the little

54

Different Kinds of Force.

obstacle will allow the exhibition of an enormous impetus. For this reason we must include in the idea of force, not only actual motion, but also the tendency to motion.

"Various kinds of force."

125. In nature we find many kinds or modes of manifestation of force; but they all agree in this, that they either actually produce, or tend to produce some sort of material motion.

The force of gravitation is, as we have seen, universal. It gives shape to our earth and all the heavenly bodies, and is the source of all our water-power, of the motion of clocks driven by weights, and of all motion due to the effect of falling or heavy matter.

It is the force of cohesion, again, that we have to overcome when we break a stone, stretch india-rubber, tear a bit of paper, saw wood, or file a piece of brass. And, though the force appears, in these cases, more as a passive or conservative one, merely resisting the tendency to motion, it is nevertheless capable of exhibiting active motive power: the reason that it does not usually appear to do so, being due to the very limited range through which the force acts. When, however, we let go the string of a bent bow, the activity of the cohesive force of the bow is exhibited in the motion of the arrow.

Chemical force operates within still more minute distances, and we cannot see immediately its manifestation as motion. But in the projection of a cannon ball by the explosion of gunpowder, we have an obvious proof that it does not differ from other forces in this respect.

The forces of heat, electricity, magnetism, and light are now considered to be all species of motion, discoverable and measurable only by the amount of movement they can produce or counteract.

"Measure of force."

126. The measure of a force is the momentum or quantity of motion it can produce in a given time, which will obviously depend both on the velocity and on the mass of the moving body.

If a single pound of matter were moving at the rate of one foot per second, it would possess a definite quantity of motion expressed by these words; if it were moving at the rate of ten feet a second, it would have ten times the quantity; and, lastly, if ten pounds of