Proposition 73

PROPOSITION 73. Problem

649. If the force is uniform and acting downwards and the medium resists according to some power of the ratio of the speed, to determine the curve AM (Fig.73), upon which the body by descending progresses along the horizontal AH at a steady rate.

Solution

Let A be the highest point of the curve, through which the vertical axis AP is drawn, and the speed by which the body progresses horizontally corresponds to the height b. The abscissa AP = x is taken, the applied line PM = y and the arc AM = s and let the speed of the body at M correspond to the height v, with which speed of the body the element Mm = ds is traversed. Hence then as ds is to dy thus the speed of the body along Mm, which is v , os to the horizontal speed b , hence there arises :

…

Let the force acting be equal to g, the exponent of the resistance equal to k and the resistance itself is equal to v m . With these in place, there arises the equation :

…

which equation, if the value

..

is put in place of v, expresses the nature of the curve dy 2 sought. Moreover let ds = pdy ; then [these relations arise] v = bp 2 and dx = dy ( p 2 − 1) .

On account of which there is obtained

…

which separated gives : [p. 350]

Therefore the construction of the curve sought is as follows : on taking then Q.E.I.

Corollary 1

650. If bv is restored in place of pp and y and x are defined in terms of v, then we have and In a like manner the arc is given by :

Moreover on taking bv in place of pp, then we have

Corollary 2

651. Because the equation has been separated, from that the solution satisfying a particular condition can be elicited by making the denominator thus it is that the speed v is constant. Hence let v = c; then …

for the inclined straight line, as we have found above(628) [There is a typo’ in the O. O. here, not present in the original ms.].

Corollary 3

652. Moreover in order that the body can progress horizontally with a given speed that corresponds to the height b, it is possible to define the height c from the equation (c − b ) = c m c . With which found, the inclination of the given straight line satisfying gk m the condition is obtained and the speed of the body c at A initially, by which it descends uniformly along the line.

Corollary 4

653. If the resistance vanishes and the body is moving in a vacuum then k =∝ and thus and

Therefore on integration the equation becomes which is the equation for a parabola, as the body projected freely describes.

Example

654. If the medium should be the most rare and hence k very great, then as an approximation : On this account there is obtained : [p. 352] From this latter equation there is as an approximation … which value substituted in the equation gives the equation sought between x and y for the curve.

PROPOSITION 74. Problem.

655. To find the curve AM (Fig.74), upon which the body is descending uniformly downwards in a medium with some kind of resistance, with a uniform absolute force directed downwards acting.

Solution

With the abscissa AP = x and AM = s let the speed by which the body descends regularly correspond to the height b. Again the uniform force directed downwards is g and the height corresponding to the speed at M is equal to v, and the m resistance is equal to v m ; hence the equation arises : …

Moreover it is the case that Mm : MN = v : b , thus the 2 equation becomes v = bds2 . Hence from this equation we dx have :

…

with which value substituted in the equation we have : On account of which it follows that [p. 353] and the applied line

From which equations the construction of the curve sought can be made. Q.E.I.

Corollary 1

656. From these three equations, if the equation is desired consisting only of x, y and s, that can be taken, from which the value of v can be most conveniently found, and with this subsequently placed in either of the remaining equations.

Corollary 2

657. Because the equation has the indeterminates separated from each other in turn, the particular solution can be obtained depending on the condition :

This is therefore : and thus … Hence the equation is satisfied by the inclined straight line, if the body is moving upon that with the given speed v .

Scholium 1

658. Because in the preceding and in this problem too an inclined straight line presents a particular solution, from that it can be understood that in a resisting medium an inclined straight line may be found upon which the body moves uniformly, as we have shown above (628). Moreover here each case of the problem is satisfied; if indeed the body advances upon the straight line with a uniform motion, then it is moving horizontally as well as vertically with a uniform motion ; then why not also be carried equally along any direction.

Corollary 3.

659. For the vacuum let k =∝ . On account of which then we have x = gv or v = gx and which equation integrated gives : and presents the rectifiable cubic parabola, as we have thus found (258).E

Example 1.

660. We put the resistance proportional to the speeds, then we have m = 12 and thus : if the start of the abscissas is taken in that point where the integral vanishes. Moreover from this equation there is produced : …

With this value of v substituted, there is obtained :

Or since put v = u 2 ; then which integrated gives: in which the value of v found before can be substituted, in order that the equation is produced between x and s. Example 2. 661. Now the medium resists in the ratio of the square of the speeds; then m = 1 and hence The integral of this latter equation is …

from which there arises :

and

On account of which it becomes :

which value substituted in the equation dx = ds b gives the equation between s and x v sought for the curve.

Scholium 2

662. As we have determined the curves in these two problems, upon which the moving body is carried uniformly either along the horizontal or down along the vertical, thus in a like manner it is possible to solve the problem, [p. 356] if the body should be progressing uniformly along any other direction; but that question, since nothing very pleasing is deduced from the solution, this I have set aside, and for the same reason I do not touch on isochronous problems about a centre for a resisting medium. Now I will apply myself to these problems, in which a certain law of the speeds is proposed, a problem not a little curious for resisting mediums, that has not been treated hitherto by anyone; which is not indeed a problem for the vacuum. Clearly the curve is sought, upon which the body reaches a given point with a maximum speed ; for in the vacuum the body moving on some given curve always obtains the same speed at the same place. [An implicit statement of the law of conservation of mechanical energy when there is no friction.