Resolution of velocity. In general the velocity of a point at x, y, z, is (as we have 2 2 {(da) " + (du)2 + (de) * } *. dt seen) ds dt, or, which is the same, Now denoting by u the rate at which x increases, or the velocity parallel to the axis of x, and so by v, w, for the other two; dx dy dz dt, we have u = W = Hence, calling a, B, y the dt dt' angles which the direction of motion makes with the axes, and we have ds dt, putting 9 = Hence u = 2 dx\? dt 26. A velocity in any direction may be resolved in, and perpendicular to, any other direction. The first component is found by multiplying the velocity by the cosine of the angle between the two directions-the second by using as factor the sine of the same angle. Or, it may be resolved into components in any three rectangular directions, each component being 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 rectdx dy dz angular, tat dt will still be the velocities parallel to the " dt axes, but we shall no longer have 'ds 2 dt 2 (d) 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 q directly or by its components, u, v, w, q cos 0 = u cos λ + v cos μ + w cos v. of velocity. Substitute in this equation the values of u, v, w already given, Resolution the axes, cos 0 = cos a cos λ + cos ẞ cos μ + cos y cos v. From the above expression we see at once that tion of 27. The velocity resolved in any direction is the sum of the Composicomponents (in that direction) of the three rectangular com- velocities. ponents 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: A B с 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. 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 components to the resultant. It is easy to see that as ds in the limit may be resolved into dr and rdə, where r and are polar co-ordinates of a plane curve, are the resolved parts of the velocity along, and dr dt do Composition of velocities. Acceleration. do dt do dt cos 0 + 28. The velocity of a point is said to be accelerated or retarded 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 velocity receives equal increments 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, sin 0, 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 interval the same as at its commencement. The average acceleration during any time is the whole velocity gained during that time, divided by the time. In Newton's notation v is used to express the acceleration in the direction of motion; and, if v = s, as in § 24, we have a = v = s. 2 Let v be the velocity at time t, dv its change in the interval or are ds " dt As St 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 a dv dt VOL. I. Sv St It is useful to observe that we may also write (by changing the independent variable) = a. dv ds ds dt we have a = d2s dt, > a Since v and it is evident from similar d2s But it is to be carefully observed that reasoning that the component accelerations parallel to the axes 1 dx 1 dy v dt ? v dt' ย dv ds 2 2 2 d2z 1 ds2 (de) = (de)2 + (dv) + (de)" - Zu 2 ρ ρ dt2 The direction cosines of the tangent to the path at any point x, y, z are 1 dz v dt Those of the line of resultant acceleration are 1 d2x 1 d3y 1 d2z 2 12 9 J dt ` f di è f dt where, for brevity, we denote by f the resultant acceleration. Hence the direction cosines of the plane of these two lines are dyd2z - dzd3y {(dyd2z — dzd3y)2 + (dzd3x – død3z)2 + (dxd3y— dyd2x)2}}' c) These (9) show that this plane is the osculating plane of the curve. Again, if Ø denote the angle between the two lines, we have sin 0 2 {(dyd3z – dzd3y)2 + (dzď3x − dxď3z)2 + (dxd3y – dyď2x)2)3 vfdt3 2 Accelera tion. Acceleration. or, according to the expression for the curvature (§ 9), d$3 v2 pvfdt3 fp Hence Again, cos Hence sin 0: = 30. The whole acceleration in any direction is the sum of Resolution and compo sition of ac- the components (in that direction) of the accelerations parallel celerations. to any three rectangular axes-each component acceleration G being found by the same rule as component velocities, that is, by multiplying by the cosine of the angle between the direction of the acceleration and the line along which it is to be resolved. 31. When a point moves in a curve the whole acceleration may be resolved into two parts, one in the direction of the motion and equal to the acceleration of the velocity—the other towards the centre of curvature (perpendicular therefore to the direction of motion), whose magnitude is proportional to the square of the velocity and also to the curvature of the path. The former of these changes the velocity, the other affects only the form of the path, or the direction of motion. Hence if a moving point be subject to an acceleration, constant or not, whose direction is continually perpendicular to the direction of motion, the velocity will not be altered-and the only effect of the acceleration will be to make the point move in a curve whose curvature is proportional to the acceleration at each instant. 32. In other words, if a point move in a curve, whether with a uniform or a varying velocity, its change of direction is to be regarded as constituting an acceleration towards the centre of curvature, equal in amount to the square of the velocity divided by the radius of curvature. The whole acceleration will, in every case, be the resultant of the acceleration, |