VELOCITY DUE TO FALL FROM CIRCLE. 111 either of the two others, viz. v2 = μr ̄1; in this case the velocity at an infinite distance is zero. -2 We see then that when a point starts from the position p at a distance r from s, and moves with acceleration μr2 always tending to s; if the velocity at starting is √(2μr ̄1), the path will be a parabola; if less than this, an ellipse with semi-major axis given by the formula pa1 = 2μr¬1 — v2; if greater, an hyperbola with semi-major axis given by the formula pa1v-2ur. The major axis of the orbit depends only on the velocity, not at all on the direction, of starting. -1 A special case of elliptic motion is that in which, the direction of starting being in the line sp, the ellipse reduces itself to a straight line. The foci then coincide with the extremities of the major axis, the eccentricity e=1, and the motion is the projection on aa' of motion in the circle with acceleration tending to a. Writing x for ap, and u for the angle acq, we have P x = a (1 − cosu), nt = u — sin u = 2 sin¬1 from which equations it may be verified with a little trouble that x2: == - n2a3. It fol lows that if from a point p in the ellipse a point be started with the velocity belonging to the elliptic motion in the direction sp, and have always an acceleration ur2, it will ascend to a point r such that sr = aa', and then return to p with the same velocity; so that the velocity at any point of the ellipse is that due to a fall from the circle rk. If we join ph and pro a duce it to cut the ellipse at q, we have ph=pr, hq + sq = aa' = sr, and therefore pq+sq=pr + sr. Hence if an ellipse be described with the foci s, p, to touch the circle at r, it will pass through q and touch the ellipse pq at that point (since both tangents must make equal angles with sq, hq). Thus all the orbits which can be described from p with given velocity touch an ellipse having foci s, p and major axis sr+pr. Or, in purely geometrical terms, given a focus, one point, and the length of the major axis of an ellipse, its envelop is the ellipse here specified. In the case of the further branch of a hyperbola described with acceleration from the focus, the velocity is that due to a fall out from the circle rk, from r to p. We have again ph = pr, and sqhq=sr, therefore When the nearer branch is described with acceleration to the focus, the theorem becomes rather more complex. If a point be started from p in the direction sp, with the velocity belonging to the hyperbolic orbit, and acceleration from s, its velocity will approximate to a certain definite value more and more closely as it gets further and further away. If we now suppose a point to approach from an infinite distance on the other side of s, with a velocity more and more nearly equal to the same value the greater the distance from s, but now with acceleration from s, this point will come up to the position (where sr aa), and there stop and go back. So that if now we reverse this process, start a point from = VELOCITY IN PARABOLIC MOTION. 113 rest at r and make it fall through infinity to the point p, it will arrive at p with the velocity belonging to the hyperbolic orbit. We have again ph=pr, sq-hq=sr, therefore pq+sq=pr+sr, or q lies on an ellipse with foci s, p touching the circle at r and the further branch of the hyperbola at q. Returning to the case of the ellipse, we know that if it is lengthened out until one focus goes away to an infinite distance, it will become a parabola. If however we send away the focus h, the circle rk, having a fixed centre s and a radius increasing without limit, will itself go away to infinity; and there will be no proper envelop of the different parabolic paths which pass through p. In a parabola described with acceleration towards the focus, therefore, the velocity at every point is that due to a fall from infinity; or, as we may say, the velocity in the parabola is the velocity from infinity. If, holding fast the focus h, we send s away to infinity, all the lines passing through it become parallel, and their ratios unity; so that the acceleration becomes constant in magnitude and direction, and we fall back on the previously considered case of parabolic motion. Since ha' is then a'k, the circle rk becomes the directrix; and we learn that in the parabolic motion the velocity at any point is that due to a fall from the directrix. The envelop of the orbits described by points starting from a given point with given velocity is a parabola having that point for focus and touching the common directrix at r. = GENERAL THEOREMS. THE CRITICAL ORBIT. Hence We have shewn that when the acceleration fur", then (n-1) v2+ url is a certain constant, c. For convenience, suppose that c = (n − 1)u2 where u is a certain velocity; then if we make r infinite, and suppose n greater than 1, ₪1-” will be zero, and we shall have v = u. u is the velocity at an infinite distance; and if the orbit has any infinite branch, u is the value to which the velocity of a particle going out on that branch would indefinitely approach. If however n is less than 1 or negative, will be zero when r is zero, and in this case. -n u is the velocity of passing through the centre of acceleration. If we draw a circle with that point as centre, and radius a determined by the equation (n - 1) u2 = μa1-”, then in the case n> 1, the velocity at every point is that due to a fall from this circle, either directly or through infinity; and in the case n < 1, the velocity is that due to a fall from the circle either directly or through the centre: it being understood that in passing through infinity or through the centre the sign of μ must be changed. Just as when n= 2 the parabola is a critical form of orbit, dividing from one another the ellipses and hyperbolas, so in general, an orbit in which u=0 is called a critical orbit. When n> 1, the velocity at every point of such an orbit is that due to a fall from an infinite distance (in this case μ must be negative, or the acceleration towards the centre); and when n<1, the velocity is that due to a fall from the centre, μ being positive and the acceleration away from the centre. In both cases n-1 = Con Since vph, we find † (n − 1) h2 p11μp2; or the orbit is of such a nature that p varies as a power of r. versely, in any curve in which p varies as a power of r, we can find the acceleration with which it may be described as a critical orbit. Now this is the case when mam cos me. For we know that the resolved parts of the velocity of P, along r and perpendicular to r respective == Consequently ly, are and rẻ. T n 2μ the sign of m is equivalent to taking the inverse curve, since it replaces r by a2: r. We subjoin a list of curves belonging to this class, observing that each is the pedal, the inverse, and the reciprocal of another curve of the series. m=2, n = 7; “lemniscate," m = −2, n=-1; rectan inverse and pedal of rectangular hyperbola. (hyperbola with perpendi cular asymptotes). gular hyperbola with centre at s; is its own reciprocal. The straight line, as we know, cannot be described with acceleration to any point out of it; and in fact the case n = 1, which the formula points to, is an exceptional From fμr, d, (1 v3) = ƒr = μr: r, we deduce1 v2 = μ log r or h2 = 2μp2 log r, which is not a curve of the kind here considered. one. t Another exceptional case is the logarithmic spiral, in which is proportional to p, and consequently n = 3, m = 0. A point started from a given position with the velocity from infinity and acceleration urs will describe a logarithmic spiral, in which the only thing that can vary is the angle at which it cuts all its radii vectores. In particular, if the point start at right angles to the radius vector, it will describe a circle. If we write z = x+iy, and (=§+ in, supposing x + iy 1 If x=logy, then y=e", and ÿ¿=¿y; hence =ỷ: y, or y: y=d,logy. |