But the areas OAB, OBD are areas traced out by OP in any two equal intervals; so that OP traces out equal areas in equal intervals. Q. E. D. 146. PROP. A point P is moving with uniform velocity u velos, (so that the line OP joining P to a fixed point O is tracing out equal areas in equal intervals); at a certain instant the velocity of P receives a certain addition of w velos, in the direction PO; it is required to prove that such an addition of velocity does not alter the rate at which OP is tracing out area. Let the point P be moving with u velos along AB. When P is at B let it receive an additional velocity w velos represented by BK, where BK produced passes through o. Complete the parallelogram BKRH; then the subsequent path of P is along BR, with velocity v velos represented by BR. Let AB=ut ft.; produce AB to D, making BD= AB; through D draw DC parallel to BK; then BC = vt ft.; so that AB and BC are the distances passed over by P in equal intervals of t seconds, just before and just after, the addition of the velocity w velos. Also AOB and BOC are the areas traced out in the same equal intervals. But the area BOC = BOD, since CD is parallel to OB; and the area BOD= AOB, for AB=BD. Therefore, the area BOC=AOB. That is, the area traced out by OP in any two equal intervals just before and just after the addition of the velocity BK, is the same. In other words, the rate at which OP traces out area is unchanged. Q. E. D. 147. PROP. The converse of Art. 146. If a point P, moving with uniform velocity u velos along AB, receive an additional velocity when it is at B, in such a manner that the rate, at which the line joining P to some fixed point o traces out area, is unchanged, then this additional velocity must be in the direction BO. R B K Let AOB and BOC be the areas traced out by OP in equal intervals of t seconds. Produce AB to D, making BD=AB. Let BH represent u velos; let BR represent the resultant velocity; then HR represents the additional velocity; and since BD AB = ut and BC = vt, HR is parallel to DC. But by hypothesis, the area OBC= AOB, and the area AOB = OBD. Therefore the area OBC= OBD. Wherefore DC is parallel to BO. But the additional velocity at B is parallel to DC and therefore is in the direction BO. Q. E. D. 148. It follows from the proposition of Arts. 146 and 147, that when the velocity of a moving point Preceives any series of additions each of which is such that it is in the direction of the line joining P at that instant to a fixed point O, then the line OP traces out equal areas in equal intervals. 149. And conversely, if the line joining a moving point to a fixed point o traces out areas at a constant rate per second, then whatever addition is made to its velocity that addition must always be in the direction which PO has at that instant. 150. This is true whether the additional velocities are large or small and whether the intervals are large or small. 151. Suppose now that a force of any magnitude acts on a particle P of mass m, so that the force is always in the direction of the line joining P to a fixed point O. Newton imagines this force to be the limit of a series of small instantaneous impulses acting on the particle at small intervals. These impulses will each be along the line joining P to 0 at the instant of its action. Accordingly these impulses will generate small additional velocities in the particle, each velocity being in the direction PO. Therefore however small and however frequent these impulses are, the rate at which OP traces out area will be unaltered by them. Hence, when a particle P is acted on by a force which always is directed towards a fixed point 0, OP traces out equal areas in equal intervals. 152. The converse is also true. If the motion of a particle P is such, that the line OP joining P to a fixed point 0, traces out equal areas in equal intervals, then whatever force acts upon P, the direction of this force must always be towards O. EXAMPLES. XXXVI. 1. Prove that a mass which is describing a circle with uniform speed, is acted on by a force whose direction tends always towards the centre of the circle. 2. Kepler observed that the line joining the centre of the sun S to any one of the planets P, traces out equal areas in equal intervals ; what is the direction of the force under which the planets are moving ? 3. Prove that a Planet moves faster when nearest the Sun than when it is furthest from the sun. CHAPTER XII. RELATIVE MOTION. 153. When we speak of the distance of a point, we must always fix upon another point from which that distance is to be measured. So when we speak of the velocity of a point P (that is, of the rate of increase of the distance of the point P), we must again fix upon some point o from which the distance which increases is to be measured. Thus the velocity of a point is a relative term only. 154. When the point O, from which the distance of P is measured, is at a distance from another point 0', then the distance of P from O' is the resultant of the distances of P from O and of O from O'. Similarly, when P has velocity relatively to 0, and O a velocity relatively to O', then the velocity of P relatively to O is the resultant of those two velocities. For example, when we speak of the velocity of a railway train we mean its velocity relative to some point on the earth's surface. The surface of the earth is itself in rapid motion. 155. When a number of points have, besides their own proper motion, a velocity common to each of them, this common velocity has no effect on the relative distances, after any interval, of the points from each other. 156. The same is true of an acceleration common to a number of points. |