Fig. 142.
Fig. 143.
212. Compound Straight Motion.—We call that motion compound which is produced by two or more forces acting upon the body. This may be straight or curved. I will first speak of the straight. If a man attempt to row a boat straight across a river, the point which he will reach will not be directly opposite to that from which he started, but below. Two forces act upon the boat: the current tending to carry it straight down the stream, and his rowing tending to carry it straight across. The boat will go in neither of these directions, but in a line between them. Let A B, Fig. 142, represent the bank of the river, from which he starts at A, with the bow of the boat pointing to C, on the opposite bank. Suppose now that in the time that it takes him to row across the current would carry him down to B if he did not row at all. He will in this time, by the two forces together, reach the point D, opposite to B, his course being the line A D. So if the wind blow upon a vessel in such a way as to carry it eastward, and a current is pushing it southward, the vessel will run in a middle line, viz., southeast. For the same reason if a boy kick a foot-ball already in motion, it will not be carried in the direction in which he kicks it, but in a line between that direction and the direction in which its former motion was carrying it. In swimming, flying, rowing, etc., we have examples of compound motion, the middle line between the directions of the forces always being taken by the body moved.
Fig. 144.
Fig. 145.
If we take Fig. 142, illustrating the movement of the boat, and draw two lines, one from A to C and the other from B to D, we shall have the parallelogram A C D B, Fig. 143, in which the line A C represents the force of the rowing, A B the force of the current, and A D the path of the boat. You see, then, that if we wish to find in what direction and how far in a given time a body acted upon by two forces will move, we are to draw two lines in the direction of these forces, and of a length in proportion to the distances to which they would move it in that time; then by drawing two lines parallel to these we shall have a parallelogram, and the diagonal of this will represent the distance and the course of the moving body. If a body be acted upon by two equal forces and at right angles to each other, the figure described will be a square, as you see in Fig. 144. If they vary from being at right angles to each other the figure will vary in the same proportion from the square figure, as seen in Figs. 145 and 146. In the three figures A B and A D represent the two forces, and A C the resulting motion. You observe by these diagrams that the nearer the two forces come to being in the same direction the farther will they move the body. You see this in the different lengths of the diagonals in Fig. 144 and Fig. 146. The more nearly, therefore, the wind coincides with the current the more rapidly will a vessel be carried along before the wind. When, on the other hand, the angle at which two forces act upon a body is much greater than a right angle, they will propel it but a small distance. Thus if two forces act on a body in the directions D A and D C, Fig. 147, they will move it only the distance represented by the diagonal D B. This diagram represents the motion of a vessel sailing almost directly against a current by a wind the force of which is equal to that of the current, while Fig. 146 represents the motion of a vessel where wind and current being of equal force very nearly coincide. In the above diagrams I have supposed the forces to be equal; but the same truth can be shown in regard to unequal forces as seen in Fig. 143.