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| {10} THEOREMA II, PROPOSITIO II. | THEOREM II, PROPOSITION II |
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| Si aliquod mobile motu uniformiter accelerato descendat ex quiete, spatia quibuscunque temporibus ab ipso peracta, sunt inter se in duplicata ratione eorundem temporum, nempe ut eorundem temporum quadrata. | The spaces described by a body falling from rest with a uniformly accelerated motion are to each other as the squares of the time-intervals employed in traversing these distances. |
| Intelligatur, fluxus temporis ex aliquo primo instanti A repraesentari per extensionem AB, in qua sumantur duo quaelibet tempora AD, AE; sitque HI linea, in qua mobile ex puncto H, tanquam primo motus principio, descendat uniformiter acceleratum; sitque spatium HL peractum primo tempore AD, HM vero sit spatium per quod descenderit in {20} tempore AE: dico, spatium MH ad spatium HL esse in duplicata ratione eius quam habet tempus EA ad tempus AD; seu dicamus, spatia MH, HL eandem habere rationem quam habent quadrata EA, AD. Ponatur linea AC, quemcunque angulum cum ipsa ab continens; ex punctis vero D, E ductae sint parallelae DO, EP: quarum DO repraesentabit maximum gradum velocitatis acquisitae in instanti D temporis AD; PE vero, maximum gradum velocitatis acquisitae in instanti E temporis AE. Quia vero supra demonstratum est, quod attinet ad spatia peracta, aequalia esse inter se illa, quorum alterum {30} conficitur a mobili ex quiete motu uniformiter accelerato, alterum vero quod tempore eodem conficitur a mobili motu aequabili delato, cuius velocitas subdupla sit maximae in motu accelerato acquisitae; constat, spatia MH, LH esse eadem quae motibus aequalibus, quorum velocitates essent ut dimidiae PE, OD, conficerentur in temporibus EA, DA. Si igitur ostensum fuerit, haec spatia MH, LH esse in {210} duplicata ratione temporum EA, DA, intentum probatum erit. Verum in quarta propositione primi libri demonstratum est, mobilium aequabili motu latorum spatia peracta habere inter se rationem compositam ex ratione velocitatum et ex ratione temporum: hic autem ratio velocitatum est eadem cum ratione temporum (quam enim rationem habet dimidia PE ad dimidiam OD, seu tota PE ad totam OD, hanc habet AE ad AD): ergo ratio spatiorum peractorum dupla est rationis temporum: quod erat demonstrandum. Patet etiam hinc, eandem spatiorum rationem esse duplam rationis maximorum graduum velocitatis, nempe linearum PE, OD, cum sit PE ad OD ut EA ad DA. {10} | Let the time beginning with any instant A be represented by the straight line AB in which are taken any two time-intervals AD and AE. Let HI represent the distance through which the body, starting from rest at H, falls with uniform acceleration. If HL represents the space traversed during the time-interval AD, and HM that covered during the interval AE, then the space MH stands to the space LH in a ratio which is the square of the ratio of the time AE to the time AD; or we may say simply that the distances HM and HL are related as the squares of AE and AD. Draw the line AC making any angle whatever with the line AB; and from the points D and E, draw the parallel lines DO and EP; of these two lines, DO represents the greatest velocity attained during the interval AD, while EP represents the maximum velocity acquired during the interval AE. (Condition 2/01-th-01) But it has just been proved that so far as distances traversed are concerned it is precisely the same whether a body falls from rest with a uniform acceleration or whether it falls during an equal time-interval with a constant speed which is one-half the maximum speed attained during the accelerated motion. It follows therefore that the distances HM and HL are the same as would be traversed, during the time-intervals AE and AD, by uniform velocities equal to one-half those represented by DO and EP respectively. If, therefore, one can show that the distances HM and HL are in the same ratio as the squares of the time-intervals AE and AD, our proposition will be proven. {210} (Condition 1/04-th-04) But in the fourth proposition of the first book (p. 157 above) it has been shown that the spaces traversed by two particles in uniform motion bear to one another a ratio which is equal to the product of the ratio of the velocities by the ratio of the times. (Condition 2/00-th-00-dialog1) But in this case the ratio of the velocities is the same as the ratio of the time-intervals (for the ratio of AE to AD is the same as that of 1/2 EP to 1/2 DO or of EP to DO). Hence the ratio of the spaces traversed is the same as the squared ratio of the time-intervals. Q. E. D. Evidently then the ratio of the distances is the square of the ratio of the final velocities, that is, of the lines EP and DO, since these are to each other as AE to AD. |
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