29. THE motion of a point in space will be completely determined, if we know the law to which its velocity is subject throughout the motion; and this is usually discovered by knowing the law of the acceleration by which it is affected. Since the propositions called the parallelogram and parallelepiped of velocities (which equally hold good for accelerations) shew that we may consider that part of the motion which results from any one irrespectively of the others, it will be convenient to resolve the velocity and acceleration into directions at right angles to each other, and to consider them separately. A. 30. We shall now find certain relations between space, time, velocity, and acceleration, here denoted by the letters s, t, v, a. ds dt We have = (Art. 15), d's ... a= if t be considered independent variable; or a dt2 ds dv dt ds dv = v if s be considered independent variable. ds' The motion has here been supposed rectilinear: if it be cur the motion of a point along a curve with that of one along the tangent, and bearing in mind that the corresponding elements of the arc and tangent are coincident. The other equations cannot be assumed to be true, because the change of the velocity is not entirely along the tangent. A. 31. If, in the enunciation of Newton's 10th Lemma, we read "point" instead of "body," and "acceleration" instead of "force," the reasoning still holds good, and we have If t be considered independent variable, this vanishing fraction, being evaluated in the usual way, becomes, after two as before obtained. d's differentiations, d A. 32. If x, y, z be the co-ordinates of the moving point at time t, the cosines of the angles which the direction of the ds velocity makes with the rectangular axes of x, y, z, are dt and therefore (Art. 10) the resolved parts of the velocity in the directions of the axes will be and therefore the accelerations in the same directions will be A. 33. It is sometimes convenient to consider the position of the moving point as determined by polar co-ordinates. and the velocity in direction PT at right angles to OP, and in direction of the increase of 0, A. 34. The accelerations may also be obtained in the * The difference in form between these expressions and those for the accelerations in x and y is owing to the fact that the directions of OP, PT are variable, whereas those of x and y are fixed. d2r That cannot represent the acceleration in the direction of OP may be condt2 cluded from the consideration of the simple case of motion where r is constant, and der ... 0: but this motion is circular, and there must be an acceleration along the dt2 radius, otherwise the point would move in a straight line; therefore when there is yet an acceleration existing towards the pole, whence the acceleration in OP as d2x day and do those in the directions of x and y. A direct dt2 demonstration of these formulæ may be found in Sandeman's treatise Of the Motion of a Single Particle. It must be borne in mind that the positive direction along the radius vector is measured away from the pole, and the positive direction at right angles to this is measured in the same direction as that in which increases. A. 35. It may be also advantageous to consider the velocity or acceleration as resolved along the tangent and normal. and the velocity in direction of the tangent is direction of the normal is 0. dt D The positive directions in this case are measured in the direction of increase of s, and in that which is considered the positive direction of curvature. In the last two sets of resolutions, we have for simplicity considered the motion as taking place in one plane. A. 36. If we know the conditions to which the motion is subject, we can equate the expressions (whether for velocity or acceleration) obtained in the preceding Articles to certain given quantities, and thus obtain sets of differential equations, the solution of which will determine the motion. As the expressions of the preceding Articles have all been derived from those in * These formulæ may be obtained very elegantly in the following manner: let the ds (which = da ). angle PTæ = 4, and let v be the velocity at P, ( for as does not appear except in a differential coefficient, we may increase it by some constant angle, so as to pass to the angle required in the intrinsic equation to the curve: the negative sign has been rejected by considering the direction of the curvature. |