Proposition 104

PROPOSITION 104. THEOREM.

860. If a body is moving in a medium with resistance acted on by some number of absolute forces, the resistive force does not disturb the action of the other absolute forces in any way, except that the tangential force arising from that is diminished.

DEMONSTRATION.

From the last chapter it has been explained well enough that all absolute forces can be resolved into two forces, the tangential and the normal, if the motion is to be in the same plane. But if the body does not move in the same plane, then three equivalent forces can be assigned in place of any number of forces acting, of which one is the tangential and two are normal. But the force that the resistance exerts on the body, is always put to agree with the direction of the body (117). On account of which the resistive force has to be referred to the tangential force that it diminishes, [p. 370] since the motion of the body is slowed down, and indeed it does not in short affect the normal forces. Therefore it is evident that the resistance has no effect on the absolute forces, except in as much as the tangential force arising from these is diminished by the resistance. Q.E.D.

Corollary 1.

861. Therefore the whole effect of the resistance is consistent with changing the speed of the body and leaves the direction unchanged, except in as much as the action of the normal force varies with the variation of the speed.

Corollary 2.

862. Therefore except besides by the aid of [normal] absolute forces, it is not possible for the body to move along a curve, but always to progress along a straight line, while it loses its motion.

Scholium 1.

863. Therefore in this chapter, in which we treat curvilinear motion, it is necessary that we consider absolute forces likewise and these are of such a kind that they can be resolved to give a normal force, and which are different from what we discussed in Chapter IV. On this account the first force we consider pulls towards a point at an infinite or the direction of this force is always kept parallel to itself. From there we progress to centripetal forces and to other forces set out in whatever manner. And hence also we submit to our analysis motions not contained in the same plane, [p. 371] of such kinds as arise from motion in a resistive medium.

Corollary 3.

864. If the tangential force is T, and either the one normal is N , or the two normals are N and M , and the force of resistance is R, then the rules containing the effect of these forces that we gave in the previous chapter are also to be applied here, except that we put T – R in place of T for these.

Scholium 2.

865. As the force of the resistance is made to depend on the speed of the body, it is necessary to be explained by a rule for the resistance, and the explanation has been widely set out in Chapter IV. Truly in this chapter a wide range of resistances are uncovered to be treated, which did not find a place in that previous chapter. Besides, this treatment thus has been subdivided, in order that at first we can determine the curve described and the motion of the body, from the given absolute forces and the resistance. Then if the curve and the absolute force is given, from these we deduce the resistance. Following this, in the third place, from the given curve and the resistance the absolute force in a given direction is to be investigated. Then from the given curve, with both the speed of the body at individual points and the resistive force, the absolute force and its direction can be found. But in the first part of this chapter [p. 372] the division of motion in coplanar and non-coplanar parts is agreed upon.

PROPOSITION 105. PROBLEM.

866. If a body is moving in a medium with some resistance and is acted upon by some absolute forces, yet thus, so that the motion is completed in the same plane, to define the rules that the body observes in its motion.

SOLUTION.

The body describes the curve AMB on account of the forces acting (Fig.81); let the speed of this at the point M correspond to the height v and the element of the curve Mm = ds. Again the normal force is put equal to N, and hence the direction of this force is along the normal MN to the curve, truly the tangential force arising from the same absolute forces is equal to T, the direction of which is MT, the tangent of the curve at M. And the force of the resistance at M is equal to R. With these put in place, the motion of the body can be defined from the normal force N and from the force along the tangent T – R (864). Now let the radius of osculation at M be equal to r and so

N = 2rv and dv = ( T − R )ds

From these two equations if v is eliminated, the equation is produced expressing the nature of the curve, and likewise the speed of the body at individual points can be observed from the equation N = 2rv . Q.E.I. [p. 373]

Corollary 1

867. Therefore there arises v = Nr . Hence there is found : dv = Ndr +2 rdN . Which value, if 2 in place of v, and placed in the equation dv = ( T − R )ds in place of dv and in R is put Nr 2 the equation for the curve described by the body is produced.

Corollary 2

868. If in R v should have a single dimension, since that comes about if the resistance is proportional to the square of the speed, then the equation dv = ( T − R )ds is able to be separated and each force can be determined from that. And this equation solved with v = Nr gives a simpler equation for the curve described. 2

Scholium

869. Besides this case, in which v has a single dimension in R, many others are given, for which the equation dv = ( T − R )ds can be integrated; but there is no need to explain these, as v is eliminated in any case. Truly we have noted the case for this particular idea, since in fact it pertains to the resistance of fluids and which therefore we will examine with more care before the others.