The Exponent of Resistance

DEFINITION 19

376. The exponent of the resistance is the height corresponding to its speed, which if the body has this speed, then the resistance experienced is equal to the force of gravity.

In this case the body is slowed down by the resistance to the same extent as it would be slowed in moving up under the force of gravity.

[The exponent of the resistance is hence a constant associated with that resistive force, indicating its strength. In certain circumstances where the resistance varies with position, then it is not a constant, as explained below. For uniform resistance it is constant.

The resistance is given as a negative acceleration of the form − ( kv ) , where v is the height the body falls under n constant unit acceleration due to gravity, corresponding to the speed whole number power.] v , and n is a

Corollary 1.

377. Therefore if the body in the resisting medium has a motion corresponding to an altitude v and this altitude v is equal to the exponent of the resistance, then [p. 157] the body progresses through the element of distance dx, where dv = – dx ; since in this case the resistive force is equivalent to the force of gravity, that we always put equal to 1, and the motion is slowed down.

Corollary 2.

378. Therefore with the law and the exponent of the resistance given, the diminution of the motion can be defined. Accordingly from the known exponent, what size of speed the body should have, in order that the resistive force is equal to the force of gravity, and from the given law of the resistance, then the ratio is known, according to which different speeds are lessened by the resistance.

Scholium.

379. The exponent of the resistance is either a constant, or a variable, or depends on the location of the body. The first mentioned happens in a medium or uniform fluid, which has the same resistance acting on bodies everywhere, if they move everywhere with the same speed. A resistive medium of this kind we will call uniform, clearly which is the same to the body in all places. Moreover the exponent of the resistance is variable in a medium or fluid that is not uniform, even if the resistance follows the same law in separate place also. For when the fluid or medium is denser, in which the body moves about, in that too the body experiences the greater resistance, which is equal to the speed of the motion [due to gravity] also. Cleary the resisting speed equal to gravity is greater in a rarer medium, less in a more dense medium. Moreover since dense and rare mediums depend of the location, it is evident [p. 158] that the exponent of the resistance, if it is variable, must depend on the location of the body.

DEFINITION 20

380. Here resistive mediums are called similar that have the same law of resistance. Truly these mediums are dissimilar which have different laws of resistance. Thus water and mercury are mediums of the same kind, accordingly as both have fluids resist in the ratio of the square of the speeds.

Corollary

381. Therefore if the resistances of similar media are different from each other, then the whole difference is consistent with the exponent of the resistance or with the density and rarity. Thus in water the exponent of the resistance is greater than in quicksilver, since this fluid is denser than that.

Scholium

382. Similar mediums with bodies with equal speeds are able to acquire different resistances, since the densities of these are different from each other, and it can be agreed to measure these densities from the resistances imparted to the bodies with a given speed. Indeed in fluids, as with the motion of the body performed in the fluid, it is taught, that for bodies with equal speeds, the resistances are in proportion to the densities of the fluids. We transfer this property to the resistance of any other mediums whatever the law governing the resistance : since other laws besides the ratio of the square of the speed [p. 159] are purely imaginary and they are accustomed to be used as exercises in analysis only.

PROPOSITION 49. PROBLEM.

383. For a body moving along the line AP (Fig. 37) in a medium with some kind of resistance, both the law and the exponent of which are known, with a given speed at the point P, to find the decrease in the speed as the element of distance Pp is traversed.

SOLUTION

With the element Pp = dx let the altitude corresponding to the speed at P be equal to v and the exponent of the resistance be equal to q. Therefore √q denotes the speed, which if the body has at P, then the force of the resistance will equal the strength of gravity, which in turn is equal to 1. On account of which, if v = q, then the resistive force is equal to 1 and dv = - dx (376, 377). Moreover let V be that function of the speed √v , by which the law of the resistance is expressed, and Q designates a similar function of √q, or Q is a quantity of that kind, which is produced if q is substituted in place of v in V. Therefore the resistance, that the body moving with the speed √v experiences, is equal to VQ . Which . Q. E. I. [p. 160] when the motion is slowed, will be dv = −Vdx Q

Corollary 1

384. Since the quantity V is a function of v and of a constant, and q likewise, and Q is either a constant or some function of x (375), the equation found dv = −Vdx can be freely Q separated. For we have dv = −Qdx , from which by integration or even by construction the V whole motion of the body along AP is known.

Corollary 2

385. Since the force of resistance is equal to VQ , the density of the medium can be found from this. Since indeed the density is measured from the resistance, that the body experiences moving with the given speed, it is necessary to substitute a constant quantity in V in place of v, with which done the resistance is found to vary as Q1 .

Therefore the density of the medium also will be as Q1 , or inversely with Q.

Scholium.

386. Here VQ denotes not only the force acting, but now the strength of the retarding resistance itself [It is now clear that Euler identifies the strength of a force with the acceleration it produces], on account of which there is no need for the mass of the body to be included in the calculation. Moreover here the mass of the body is constant or we put the masses of many bodies equal to each other. Indeed I have agreed not to deliberate about this here, which will only come to be used in a single case, except to extend the necessity to return to more complicated cases. [p. 161]