In general the velocity of a point at x, y, z, is (as we have Resolution of velocity. Now dt dr =v (suppose) is the rate at which x increases, or the velocity parallel to the axis of x, and so of the other two. Hence if a, ẞ, y be the angles which the direction of motion makes with Velocity. Resolution Space described in one second, Ten times the space described in the first tenth of a second, A hundred hundredth and so on, give nearer and nearer approximations to the velocity at the beginning of the first second. The whole foundation & the differential calculus is, in fact, contained in this simple question, "What is the rate at which this space described increases?" i.e., What is the velocity of the moving point? Let a point which has described a space s in time & proced to describe an additional space ds in time ôt, and let v, be the greatest, and v, the least, velocity which it has during the interv St. Then, evidently, vast, ds <v1dt, ds> But as St diminishes, the values of v1 and v, become more and more nearly equal, and in the limit, each is equal to the velocity at time t. Hence 2= ds dt . 25. The above definition of velocity is equally applicabl of velocity, whether the point move in a straight or curved line; but, since in the latter case the direction of motion continually changes the mere amount of the velocity is not sufficient completely to describe the motion, and we must have in every such case additional data to remove the uncertainty. In such cases as this (we may explain once for all) the method commonly employed, whether we deal with velocities, or as we shall do farther on with accelerations and forces, con sists mainly in studying, not the velocity, acceleration, or fore directly, but its resolved parts parallel to any three assume directions at right angles to each other. Thus, for a train moving up an incline in a NE direction, we may have given the whole velocity and the steepness of the incline, or we may express the same ideas thus-the train is moving simultaneously northward, eastward, and upward-and the motion as to amount and direction will be completely known if we know separately the northward, eastward, and upward velocities-these being called the components of the whole velocity in the three mutually per pendicular directions N, E, and up. In general the velocity of a point at x, y, z, is (as we have Resolution of velocity. dr Now = (suppose) is the rate at which x increases, or the dt Hence velocity parallel to the axis of x, and so of the other two. per 26. A velocity in any direction may be resolved in, and pendicular to, any other direction. The first component is formed by multiplying the velocity by the cosine of the angle between the two directions-the second by using as factor the ine of the same angle. Or, it may be resolved into components in any three rectangular directions, each component -ing formed by multiplying the whole velocity by the cosine. of the angle between its direction and that of the component. It is useful to remark that if the axes of x, y, z are not rectanwill still be the velocities parallel to the axes, gular, dx dy dt dt dz dt We leave as an exercise for the student the determination of the correct expression for the whole velocity in terms of its components. If we resolve the velocity along a line whose inclinations to the axes are λ, μ, v, and which makes an angle with the direction of motion, we find the two expressions below (which must of course be equal) according as we resolve v directly or by its components, Substitute in this equation the values of v, vy, vz already given, § 25, and we have the well-known geometrical theorem for the angle between two straight lines which make given angles with the axes, costcosacosλ+cosẞcosu+cosycosv. From the above expression we see at once that Composi tion of 27. The velocity resolved in any direction is the sum of the velocities. resolved parts (in that direction) of the three rectangular components of the whole velocity. And, if we consider motion in one plane, this is still true, only we have but two rectangular components. These propositions are virtually equivalent to the following obvious geometrical construction : B To compound any two velocities as OA, OB in the figure from A draw AC parallel and equal to OB. Join OC:-then OC is the resultant velocity in magnitude and direction. A OC is evidently the diagonal of the parallelogram two of whose sides are OA, OB. Hence the resultant of velocities represented by the sides of any closed polygon whatever, whether in one plane or not, taken all in the same order, is zero. Hence also the resultant of velocities represented by all the sides of a polygon but one, taken in order, is represented by that one taken in the opposite direction. When there are two velocities or three velocities in two or in three rectangular directions, the resultant is the square root of the sum of their squares-and the cosines of the inclination of its direction to the given directions are the ratios of the com ponents to the resultant. It is easy to see that as ds in the limit may be resolved into & and rôe, where r and are polar co-ordinates of a plane curve, are the resolved parts of the velocity along, and perpendicular to, the radius vector. We may obtain the same result de dr and r dt 28. The velocity of a point is said to be accelerated or re- Acceleration. tarded according as it increases or diminishes, but the word acceleration is generally used in either sense, on the understanding that we may regard its quantity as either positive or negative. Acceleration of velocity may of course be either uniform or variable. It is said to be uniform when the point receives equal increments of velocity in equal times, and is then measured by the actual increase of velocity per unit of time. If we choose as the unit of acceleration that which adds a unit of velocity per unit of time to the velocity of a point, an acceleration measured by a will add a units of velocity in unit of time and, therefore, at units of velocity in t units of time. Hence if V be the change in the velocity during the interval t, V = at. 29. Acceleration is variable when the point's velocity does not receive equal increments in successive equal periods of time. It is then measured by the increment of velocity, which would have been generated in a unit of time had the acceleration remained throughout that unit the same as at its commencement. The average acceleration during any time is the whole velocity gained during that time, divided by the time. Let v be the velocity at time t, dv its change in the interval St, a, and a the greatest and least values of the acceleration during the interval ôt. Then, evidently, As &t is taken smaller and smaller, the values of a, and a, approximate infinitely to each other, and to that of a the required acceleration at time t. Hence dv ds It is useful to observe that we may also write (by changing the independent variable) do etc. |