The law of the resistance

Table of Contents

DEFINITION 18

The law of the resistance is the force or function of the speed of the body, to which the resistance is in proportion. Thus if the resistance is proportional to the square of the speed, the law of the resistance is the square of the speed.

Corollary 1

Therefore it is known from the law of the resistance, if many equal points are carried forwards with different speeds, how their motions diminish between themselves. And from the given decrease in the speed of one point the decreases in the speeds of the rest can also be found.

Corollary 2

If therefore from one step in the speed given for the ratio of the resistance to the force of gravity, for all the other steps too the ratio between the resistance and the force of gravity can be found from the law of the resistance. And from this the effect of the resistance on the motion of the body can be found.

Scholium 1

The strength of the resistance extends as far as the general idea of force and thus it is homogeneous with the force of gravity, as this will become apparent when the motion of bodies in fluids is treated.

Therefore an absolute force can always be assigned to produce the same effect on the body as resistance. Truly this absolute force will always depend on the speed of the body, since on this account, in the expression of this, the speed or the height corresponding to the speed will be present.

Therefore in this way the motion of the body in a medium with resistance is reduced to the motion of a body acted on by an absolute force, since the laws will have been set out as in the second chapter above, from which all the questions can be resolved. [Thus, one assumes that the medium is at rest, so that absolute forces are acting.]

Scholium 2

The direction of the force of the resistance in this tract of ours will always be in agreement with the motion of the body and in the opposite direction (117). On account of which the absolute force to be substituted for that will always be one of retardation, with the direction of the motion not changed. And thus it is evident that the expression for the resistive force is made negative as often as the body moves in a contrary direction, and that motion is one of being accelerated. Indeed it is not possible to have this case for a body at rest at a point in a fluid, but yet in the calculation this often occurs for the given motion of the body in the fluid investigated.

Corollary 3

Therefore the motion of a body in a fluid with resistance, if not acted on by any other force, must be to move in a straight line. For since the direction of the motion is not changed by a resistive force, the motion of the body, that naturally peruses a straight line, must always by necessity be made in the same straight line.

Corollary 4

If in addition an absolute force is acting, the direction of which is always in the same direction as the motion, then the body in the resistive medium also proceeds in a straight line. For neither will this absolute force nor the resistive force change the direction of the motion.

Scholium 3

Therefore in this chapter, in which we have only established for rectilinear motion to be explained, we do not join other absolute forces with the resistive force, except when their directions agrees with the direction of the motion. On account of this, it is possible to consider all the forces that we have used in the preceding chapter with resistance to be included. Moreover before we go forwards to absolute forces, it will be convenient to put the motion of bodies impeded by a single resistive force under scrutiny, from which it will be easier for us to progress from simpler to more complex situations.

Scholium 4

In the expression of the law of resistance or in that function of the speed, besides the height corresponding to the speed v there can be constant quantities present, but we exclude all variable quantities depending on the motion of the body. Indeed it may happen that the resistance experienced by a wide body with equal speeds, shall be greater or less as that body goes from one place to another; as happens when the fluid in which the body moves is denser in one place and indeed rarer in another, in which case it is necessary to take into account the position in the expression for the resistance. Yet neither is it appropriate to consider the position in the law governing the resistance we wish to express, for in that account of the resistance when the body is put in the same place with various speeds. Truly the distinction that can arise from the variation in the position, we understand to be in the exponent of the resistance, which likewise indicates the intensities of the resistances. [By intensity Euler means the force per unit mass or the contribution to the acceleration or deceleration associated with the force acting on the body.]