MECHANICS 681 And it is easy to see from these expressions that which gives the speed acquired in terms of the space traversed. ^tone 29. This is the only case in which the result can be aoving reached without formally using the methods of the integral erti " calculus. These expressions enable us at once to solve a a ^ great number of simple questions connected with the motion of a stone or bullet, under the action of gravity, in a vertical line. For it is found by experiment that gravity impresses, in every second, a downward speed of 32 - 2 feet per second on an unsupported body; and, by the Second Law, this is independent of the body s previous motion. Hence, if a stone be let fall, its speed after t seconds is 32 2#, and the space fallen through is 16 lt 2 . Also, if it fall through s feet, it will acquire a speed whose square is v 2 = 6 4 4s. Again, if a stone be thrown upwards with a speed of 300 feet per second, after t seconds its speed will be 300 - 32 2, and the height to which it has then ascended is 300<- 16 li! 2 . Thus it stops, and turns, after g^ 300 2 feet. seconds ; and the greatest height it reaches is From the statement above, putting s = v, we find . T v.-, - v, dv S = V = Li ~ = < 2 <} dt From this expression the preceding results may be at once obtained. Thus, assuming we have by integration and again s = s + ~Vt + |a< 2 . As an instance of the indirect problem i.e. , to find the speed, and its rate of increase, when the law of the motion is given: suppose (This equation describes the simplest form of vibratory motion, and will be fully treated later.) We have, by taking the fluxion, s= -ftwsinwt, and again s= -&rcosco< = u"s . 30. Velocity, as we have already said, involves the ideas of speed and of direction of motion conjointly. 1 To compound two velocities (as is required in the application of Newton s Second Law), we have the following obvious construction. From any fixed point (fig,. 2), draw a line OA representing, in magni tude and direction, one of the two velocities. From its extremity A draw AB repre senting in the same way, and on the same scale, the other. Complete the triangle OAB. Then OB represents, in magnitude and direction (still on the same scale), the resultant velocity. We have called the construction obvious because one has only to think of hoiv a point can be said to have simultaneous velocities, in order to see its truth. Thus, if OA repre sents the velocity of a railway train, AB that of a pas senger walking in a saloon carriage, O may be looked upon as the position of the point of the carriage at which he began his walk, at the moment when he did begin it; while B represents the position of the point of the carriage which he has reached at the end of his walk, just at the 1 It is, in fact, in the language of quaternions, a "vector," of which the speed is the "tensor "or length, and of which the "versor" assigns the direction. And the laws of composition of velocities are in all respects the same as tho.se of vectors. A Fig. 3. moment when he did reach it. Here OA is the velocity of the carriage relative to the earth, AB that of the pas senger relative to the carriage. This proposition may be called the triangle of velocities. Another obvious mode of stating it is to complete the parallelogram of which OB is a diagonal (fig. 3) ; and then we have the same construction in the form : If the two velocities to be compounded, represented by OA and OC, be taken as con tiguous sides of a parallelogram, the conterminous diagonal OB represents their resultant. 31. From the triangle of velocities we may pass at once Composi- to the polygon of velocities, which gives us the resultant of tion of any number of simultaneous velocities. Thus, beginning velocities - as above at any point (fig. 4), lay off OA, AB, BC (however many there may be) as successive sides of a polygon all taken in the same direction round. The separate velocities may be in one plane or not. When this is done, the final pint C is easily seen to be independent of the order in which the separate velocities were taken, and is thus a perfectly definite point. OC, completing the polygon, represents the resultant velocity. But it is taken in the opposite direction round. If C coincide with O, there is no resultant ; i.e., a point which has, simultaneously, velocities represented by the successive sides of any polygon, all taken the same way round, is at rest. In what precedes, we have denoted the position of the moving point in its known path by the single quantity s. But if we think of its Cartesian coordinates x, y, z, we see that in general each of these must vary during the motion. And just as we represented the whole speed by s so we may speak of x as the speed in the direction of the axis of x, &c. And now we have a hint of a most important character. For, by the ordinary laws of the differential calculus, we have three equations of the form dx- dx -- or dx. Now is the cosine of the inclination of the tangent at s to ds the axis of x. And so with y and z. These give us Hence we see that a speed in any direction may be resolved into three in any assigned directions at right angles to one another ; that the speed in any one of these is determined by multiplying the whole speed by the cosine of the angle between its direction and that of its resolved part ; and that the square of the whole speed is the sum of the squares of the speeds in the resolved motions. These results, however, can be obtained more directly, and in a more instructive manner, by the consideration of " velocities," and not of mere "speeds." But, before we take this step, let us take the second fluxions of the coordinates, and see to what they lead us. From we obtain at once dx . -r ds d-x or, introducing in the last term, both as a multiplier and as a divisor, the radius of curvature of the path, dx .. cPx i 2
- = -7- s + P~r^ >
as da- p with similar expressions for y and z. These show that the rates of increase of speed, parallel to the three axes respectively, may be considered as made up of the resolved parts of the two directed quantities S and s 2 fp . The first is in the direction of the tangent to the path, the second in the direction of the radius of curvature ; and the law of resolution is, for each, multiplication by the cosine of the angle between the two directions concerned. We shall presently recognize these as the components of the acceleration.
XV. 86