Proposition 32

Proposition 32. Problem

264. Let C be the centre of the forces (Fig. 25) that attract bodies in some ratio of the distance, and by this force a body at rest at A is drawn forwards; the speed of this body is then sought at any point in the interval AC. [p. 104]

SOLUTION

Let AC = a, AP = x; and the speed that the body has at P is that corresponding to having fallen from the height v. The attraction shall be given as the ratio of the distance raised to some power n, and f is taken as the distance from C, at which the force on the body towards C is equal to the weight of the body, if it should be placed on the surface of the earth.

Therefore the strength of the acceleration, by which the body at P is pulled towards C, will be as the strength of gravity, that I put equal to 1, as CPn, i. e. as nn fxa to)( − ; on account of which the acceleration is expressed …

Therefore by taking Pp = dx then )( …

For dv is equal to dx multiplied by the strength of the acceleration (213). This integrated equation produces )1( …

For the constant C to be …

defines put x = 0, in which case by hypothesis it must become v = 0; therefore …

It is therefore found that )1( ….

Or by putting a – x = CP = y it becomes ….

From which equation the speed of the body at any point of the interval AC is known. Q. E. I.

[Thus, the force on the body at position a – x is given by some function n xak )( − , and the acceleration is mxak n /)( − , where m is the mass of the accelerated body and k is a constant of proportionality. However, when the body is at f, it is considered to have the force equal to its weight acting on it, which is just m, as g is taken as equal to 1; hence

1.mkf n = , or n fmk /= , giving the force as nn fxam /)( − , and the acceleration as nn fxa /)( − .

In addition, for motion under gravity 2 2 Vvg = , where V is the final speed at the point and g is taken as 1, then VdV = gdv and also adxVdV = for the motion under the new force; hence adx = dv, where a is the acceleration under the new force, and the point masses under gravity and under the new force have the same speed and increment in the speed; as these are not in proportion in general, each point must correspond to a different release point. This gives the ratio of the accelerations under gravity and the force, which is a function of the distance. From which it follows that )(

..

which can then be integrated as above. We would now proceed in a slightly different manner , and set the acceleration ;// dxVdVdtdV = in which case … etc.

We may note also that this first integral is in fact just the conservation of energy, as the sum of the kinetic and potential energy is related to the constant C. This relation obviously breaks down when infinite quantities are involved.]

Corollary 1

265. If n + 1 is a positive number, yn + 1 vanishes when y = 0. Therefore in this case the altitude corresponding to the speed, that the body has on arriving at C, will correspond to )1( ..

But if n + 1 is a negative number, yn + 1 will become infinitely large when y is made zero : hence in this case the body on arriving at C will have an infinitely large speed.

Corollary 2

266. But if n + 1 = 0 or n = -1, the value found from the equation itself may not be known on account of the numerator and the denominator vanishing. Because of this, it will be necessary to repeat the differential equation. Moreover, it follows that … the integral of which is )( xaflCv −−= . And it must be that C = fla, on account of which )]log( …

Which is the true value of v, when n has the value - 1, i. e. when the centripetal force varies inversely with the distance from the centre of the force.

Corollary 3

267. Therefore in this case, n = – 1, when the body arrives at the centre C, its speed is infinitely great; for it shall be that v = fl ∞ . This infinite step is to be deplored, and if a nearby value is taken, it is finite ; however if n + 1 should exceed zero a little, then the speed at C suddenly becomes finite.

Corollary 4

268. Moreover since n + 1 should be a positive number, since then the height corresponding to the speed at C is )1( .. then the speeds of many bodies falling towards the centre C, and which they have at C, are as 2 1+n a , i. e. as the 2 1+n power of the distances from which they have began the fall.

Scholium 1

269. Moreover, after the body arrives at C from A in C, where then it shall keep on moving forwards, it is not so easy to be defined.

If y is made negative in the expression found, the height corresponding to the speed at Q should be emerging ; which if it is positive, then the body again returns to Q ; but truly if it is negative, from the evidence, this body never reaches beyond C into the region CQ.

In truth this way of continuing the motion is not always possible to be adhered to; often indeed the hypothesis itself, by which the attractive force is placed before and beyond C towards the centre is opposite. In as much as the body proving to be at P, since it is being pulled down, when it arrives at Q, it is pushed up by an equal force, if CQ = CP.

This force, on account of the nature of the force which is acting on the body at Q, is negative with the former ratio and thus is to be expressed by a negative quantity.

Therefore the force at P expressed by … or … )( − must be the negative of itself, when – y is put in place of y, and that never happens, unless n is either an odd number or a fraction, of which the numerator and denominator are uneven.

Therefore for these cases the value of v is produces, when the body arrives at Q ; always in the remaining cases, since in calculating the force acting on the body at Q when indeed with the value not in agreement, the quantity elicited for the letter v is not the true value [p. 107]. If indeed n is an even number, the attracting force at Q by making y negative is equal to the force at P and clearly falls in the same place. From which it shall be, that as the body crosses the centre C on the line CQ must continue to fall to infinity, that calculation also makes clear.

Because when it disagrees with the hypothesis, it is seen that in these cases the motion of the body, after it has arrived at C, cannot be defined by the formula defined.

Moreover it is seen to be more absurd, when 2 1=n or another fraction of this kind, which changes yn into an imaginary quantity with –y put in place of y; because that may indicate that that not only is the body not attracted to C, but the force of attraction also becomes imaginary, which is indeed not possible to understand.

Corollary 5

270. Therefore if n is an odd number, the value of v itself, which is )1( .. , does not change with – y put in place of + y, since the even number n + 1 of y avoids the [sign change in the] exponent.

From which it is apparent that the speed of the body at Q is equal to that which it had before at P, if indeed CQ = CP. Therefore the motion is equal in the manner in which the body recedes in the direction CQ, by which before it approached along AC; and it shall reach as far as B, thus in order that CB = AC, where it loses all its speed.

Thus it reverts again in the same way to C, and then it arrives at A again. Which reciprocal motion, unless decreased by friction, will be carried out indefinitely.[p. 108]

Corollary 6.

271. Nevertheless the case when n = – 1, since – 1 is an odd number is to be undertaken.

For with y made negative y aflv −= , which is an imaginary amount. From which it is seen that the body never goes beyond C. Hence another judgement is seen to be brought down, when n is a negative number, even if it is odd. For a similar example of this kind occurs beyond, if n = – 3 (355).

Scholium 2

272. This is seen to be less in agreement with the truth; for the reason is hardly apparent why the body with its infinite speed that it acquires at C should be about to progress into CB rather than another region, especially when the direction of this infinite speed should follow into this region.

But whatever it shall be, here the calculation rather than our judgement being trusted and established, the jump if it is made from the infinite to the finite, is not thoroughly understood. Moreover, this opinion is further confirmed by a similar example for which a full explanation is given below, (665), if n = – 2; for in this case the speed of the body arriving at C is also infinite and directed along CB ; by no less truth, in this case the body does not progress beyond C, but suddenly reverts from C equally and approaches towards A. From which it is understood, that as often as an infinite speed should arise at C, judgement about the further motion of the body should be suspended. So for the time being only this shall be done, until we come to considering motion along curves.

With these indeed which are rectilinear, and [the resolution of this problem is] clearly connected to these(762). For neither then is the calculation which is put in place subject to this inconvenience, as it is in disagreement with the hypothesis ; but whatever is put equal to the centripetal force is not in opposition to the calculation.

Scholion 3

273. But always, when the speed at C is not infinitely great, because that happens, when the size of the number n + 1 is positive, the whole motion of the body is known by our judgement, even if the calculation is insufficient. For if the speed at C is finite and has the direction along CB, that by necessity is should have, then it may not be possible to happen, as no motion can be continued along CB. But in a like manner the motion may be continuing to recede from C, when before it was approaching along AC, and at some point Q it has the same speed that before it had at the point P placed at an equal distance from C, thus as can be understood from § 251.

Therefore the motion occurs perpetually between A to B and back again, and in returning the body completes the motion.