Proposition 52

PROPOSITION 52. PROBLEM

408. A body that is moving in some medium with resistance, is acted on by some absolute force; to determine the increase or decrease of the speed, while it runs through any element Pp (Fig. 39).

SOLUTION

Let the speed of the body at P correspond to the altitude v and the element to be traversed be Pp = dx. Again let the absolute force or rather the accelerating strength of this at P = p and the exponent of the resistance is equal to q. V designates that function of v, to which the resistance is proportional, and Q is such a function of q, as V is of v.

From these put in place the body is slowed down, as it is moved through the element Pp, by the strength of the resistance VQ (383); meanwhile likewise it is accelerated by the absolute force [i.e. acceleration or force per unit mass] p. On account of which the body is accelerated in travelling through the element Pp by a strength of force equal to p - VQ . From which it follows that dv = pdx − VQ dx . Q. E. I.

[As expressed earlier, the reader may wish to consider this in modern terms as a work – energy equation : the work done per unit mass against a constant gravitational field of intensity 1 is equal to the work done per unit mass by the applied external force from which is taken the work done against the frictional force, the constant of proportionality of which is 1/Q, also per unit mass. Euler of course could not be completely familiar with this interpretation, although Daniel Bernoulli used similar expressions in his work, corresponding to the modern idea of potential energy. This was highly convenient for Euler, as he could obtain a differential equation in terms of distances only, and bring in the time later via the known speed differential.]

Corollary 1.

409. If therefore it is the case that p > VQ , then the speed of the body travelling through the element Pp is increased; but truly for p < VQ , the speed of this is diminished. And if it is the case that p = VQ , then the speed is neither increased or decreased, but remains unchanged in traversing the element Pp.

Corollary 2

410. If the absolute force were contrary to the motion and retarding it, then dv = − pdx − VQ dx . Therefore in this case the body is retarded by each force.

Scholium 1

411. If the absolute force pulls the body downwards, according to the solution of the problem we have put in place, and the body is moving up, both the absolute force and the resistance force are in the contrary direction. Therefore this equation is then obtained : dv = − pdx − VQ dx . From which it is apparent that the motion of the ascent is not the same as the motion of the descent, since the strength acting in the ascent is not in the negative ratio of the force acting in the descent. When therefore the ascent is the same as the descent and the motion in both cases has the same speed at the same point, it is required that the strength of the resistance in the ascent is changed into a propelling force. With which done we have the equation : dv = − pdx + VQ dx , from which equation it is evident that the ascending body through Pp is to be retarded just as much as it was accelerating in the descent. [p. 172]

Scholium 2

412. The equation found : dv = pdx − VQ dx with the sum extended, on account of the defective analysis, can neither be separated or constructed; and because of this the speed of the body at P cannot be determined. Therefore a much shorter time, in which the distance AP is completed must be assigned. Hence it is necessary to abandon this general equation and to examine particular cases of descents and ascents in which the equation can be separated and the speed defined. There are three ways in which the separation of the equations in the unknowns x and v can be admitted. The first of these is, if x does not have more than one dimension. The second, if v only maintains a single dimension. The third case occurs, if x and v likewise everywhere make a number of the same dimension, or if the equation can be reduced to the aforesaid property.

Corollary 3

413. In the first case therefore we have, if p and q are constants; for then, since Q is a Qdv constant, there is produced dx = pQ −V , in which the indeterminates [or unknowns] are separated from each other in turn. Besides truly also the equation can be separated , if we = dx , which, since V depends on v put p = QA . Then indeed the equation becomes : Adv −Q Q and Q on x , it is possible to construct a solution.

Corollary 4

414. When v has a single dimension, it is required that V = v, in which case also Q = q, and [p. 173] the equation of the general form will be changed in this case , which allows the separation of the unknowns. : dv = pdx − vdx

Corollary 5

415. From which it is apparent, when a homogeneous equation will soon be formed, let V = vα and q = x β ; then Q = x αβ . Again let p = xγ , and v is computed with δ dimensions, since x has one dimension. With these put in place, that equation is changed into this : dv = xγ dx - v αβdx , in the second case γ + 1 dimensions and in the third α x αδ + 1 − αβ . Therefore we must have δ = γ + 1 and γ + 1 = αγ + α + 1 − αβ or γ ( α − 1 ) = α ( β − 1 ) . Therefore as often as we have the ratio α − 1 : α = β − 1 : γ , so also the equation can be reduced to the homogeneous form and therefore that speed determined.

Scholium 3

416. In place of V, q, and p no other functions are permitted to be formed except the powers of v and of x themselves. For, since a power of x cannot be present in V and neither q nor p can enter v, and above the number of the dimensions of x and v themselves everywhere must be the same or be able to be reduced to the same, in place of these quantities by necessity the powers ought to be assumed. On this account, I have put V = vα , q = x β and p = xγ and I have extracted the above ratio α − 1 : α = β − 1 : γ . I have neglected certain coefficients, which can safely be added in this reduction, for with these the homogeneity cannot be disturbed. Thus I can put q = Bx β and p = Cxγ , with the same ratio kept. [p. 174] For x truly not only the distance traversed AP can be substituted, but increased by some other constant, provided the differential of this is dx or some

α multiple of this. But it is not necessary to add the coefficient V = v , since by V only, the ratio of the resistance is indicated.

Corollary 6.

417. If the resisting medium is uniform medium and thus β = 0, then α − 1 : α = −1 : γ . Hence γ = 1α-α . Whereby if the law of the resistance is vα , then the absolute force must α be equal to Bx 1-α , where the equation defining the speed can be reduced to the homogeneous case.

Scholium 4

418. Thus we are about to explore these rectilinear motions in resistive mediums, as first we are to put in place an absolute constant force, and then we progress to several centripetal forces. And with these explained we contemplate inverse questions, as we did in the preceding chapter, and from the given properties of the motion so we elicit the absolute force and then the resistive force. [p. 175]