Chapter 6e

Proposition 109

Motion is the translation of a body from the place it occupies to another place. True rest is a body remaining at the same place.

Euler
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PROPOSITION 109. PROBLEM

  1. If the body is everywhere attracted downwards equally, and it is projected along the horizontal direction at A (Fig.83) with a given velocity in a uniform medium that offers resistance in the simple ratio of the speed, then to determine the curve AM that the body describes and to find the motion of the body on this curve.

SOLUTION

Since this proposition is a special case of the preceding, all the derivations remain as before. Moreover y or the applied line PM becomes negative, since the curve AM falls below AP, and we have μ = 0 and ν = 1. Therefore with the force g acting, with c the exponent of the resistance, with b the height corresponding to the speed at A and AP = x and AM = s, this differential equation is found for the curve AM : (888) and this integral: (888). And the time, in which the arc AM is traversed, is equal to (890). Which equations determine the curve AM, and also the motion on the curve. Q.E.I

Corollary 1.

  1. If l 2 bc is converted into a series, it gives : 2 bc − x On account of which, we have [p. 385] and the time, in which the arc AM is traversed, is equal to

Corollary 2.

  1. In a vacuum therefore, when c is made infinitely great, the equation becomes gx 2 y = 4b ; in which case therefore the curve AM goes into a parabola, the parameter of which is 4gb , and the time in which the arc AM is completed is equal to x and b gx 2 v = b + 4b = b + gy , as it is clear to recollect from Prop. 72 (564).

Corollary 3

  1. By taking AE = 2 bc the vertical line EF is the asymptote of the curve AM. Whereby the perpendicular MQ is sent from M to EF, and we have MQ = PE = 2 bc − x and EQ = y . On putting MQ = z, we have and the time in which the arc AM is traversed is equal to 2 c l 2 zbc .

Corollary 4

  1. Therefore the point E through which the asymptote EF passes is as far as the body that has been sent from the point A can reach, if there is no aid to the force g acting, before all the motion has been made available. And likewise in a similar way it is apparent that the time to pass through AM is equal to the time to pass along AP with the force g vanishing. [p. 386] Hence it is understood from this that g is not present in the expression for the time.

Corollary 5

  1. The tangent MT drawn from M is given by Through M draw MR making an angle with EF, the tangent of which is equal to b . g c With which accomplished, QR = gz c and thus RT = 2 gc. b

Corollary 6

  1. Therefore if MR is considered as an oblique applied line [y-axis] of the curve AM to the axis EF, then on account of the constant sub tangent RT, the curve AM is logarithmic with the oblique-angled sub tangent equal to 2gc, and the tangent of the angle of inclination of the applied line MR to the asymptote EF = angle MRQ = z/RQ = b . [i. e. the tangent of the g c b , which is constant, and hence MR can be considered as an g c oblique axis; as this axis slides along EF as the point M varies, the length RT remains constant. A curve that has, no doubt, other interesting properties.]

Scholium

  1. The trajectory in this resisting medium and with the force acting under this hypothesis can be constructed not only with the aid of logarithms, but has been examined by Johan Bernoulli in Act. Erud. Lips. 1719 to be an oblique-angled logarithmic curve, the solution of which agrees uncommonly well with our solution. (See note 2 above.)

PROPOSITION 110. PROBLEM

  1. With the absolute uniform force put acting along the vertical direction MP (Fig.82) and the medium, that is also put as uniform, [p. 387] with the resistance in some ratio of the multiple of the speeds, to determine the curve AM described by the projected body. SOLUTION. With AP = x, PM = y, Mn = ds, the speed at M = v , and the force equal to g as before, with the exponent of the resisting medium equal to c, with the medium resisting in the ratio of the 2m th power of the speeds, and given by m R = v m and P = g (870). Whereby these equations are to c be had : from which the curve AM as well as the motion of the body on the curve can be gds 2 determined. Moreover, the equation v = − 2ddy gives Hence with v eliminated, this equation is arrived at for the nature of the curve : For the construction of the curve, put dy = pdx and there arises : From which on substituting there becomes dp Again put dx = q and then

Hence there arises and on integrating [p. 388] From which equation q is given in terms of p, from which it is found on taking the abscissa x = dp pdp ∫ q , there corresponds the applied line y = ∫ q . And with the height corresponding to the height and the time in which the arc AM is completed, i. e. ∫ dsv , is equal to Q.E.I.

Corollary 1.

  1. It is evident that whenever 2m is either a positive or negative odd number, the value of q can be shown algebraically in terms of p. Corollary 2.
  2. If the resistance is constant or m = 0 and the body initially at A is projected with a speed b along the horizontal AP, the applied line PM or y likewise taken as negative, has these equations Hence this equation is produced : Corollary 3.
  3. Moreover this case is easier to handle if m = 0 in the differential equation of the 2ddy 2 third order, for it produces gd 3 y = − ds or on substitution in terms of p and q this equation is made : the integral of which isEULER’S MECHANICA VOL. 1. Chapter Six (part a). Translated and annotated by Ian Bruce. page 562 or from the noted homogeneity [p. 389] Hence there arises Which again on integration gives : ∫ And hence there is found y = pdx and this is completed on integration to give : Therefore it is apparent that this curve is algebraic only if g is 1 or 2.

Scholium

  1. Equally general to our solution is the solution given in the Acta Erud. Lipt. in May 1719 by Johan Bernoulli1 on trajectories in resistive media, where the general construction of these curves was given. But before we leave the constant force hypothesis, we must solve the inverse problems, in which we determine the resistance that is effective, in order that the body describes a given curve acted on by a constant downwards force according to the hypothesis. For this matter has been treated several times, first by Newton2 in the Phil. Princ. then again by Johan Bernoulli3 in the Act. Lips, A, 1713: where the sharpest of men have noted many interesting things. [p. 390] [Thus, from the observed curve, one can determine whether or not the resistance has a particular form.]

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