Proposition 28

Table of Contents

Proposition 28 Theorem

For a body falling through the distance AP (Fig. 22), as we have put in place as hitherto, the [final] speed at P is of such a size, that if it progressed uniformly at this speed for the same time in which the body had fallen through AP, then it would be able to complete a distance twice as great as AP.

DEMONSTRATION

With everything kept, that we put in place in the preceding proposition, with the body A, the force g, with the distance described x, with the speed acquired at P v and with the time of the descent t, then g ..

and A gx v = (206).

Then on account of this we have … and thus …

But this expression also gives the time, in which the distance 2x will be traversed with a uniform speed of v , since v x2 is divided by 250, and then we find the number to be expressed in seconds (220). Consequently the distance 2x is travelled in the same time with the speed v , in which the distance x is fallen under a uniform acceleration. Q. E. D. [Thus the final speed is twice the average speed for the motion of a body released from rest under gravity.]

Corollary 1

Therefore a body acted on by a uniform force falling in a time t through a distance x will acquire as great a speed, as that by which a body can progress uniformly through the same distance in half the time t.

Corollary 2

Since on the surface of the earth bodies are falling in a time of one second a distance of 15625 scruples of Rhenish feet, their final speed acquired in the fall will be as great as that with a uniform motion over a distance of 31250 in one second, or 15625 scruples traversed in a time of half a second. [p. 92]

Corollary 3

When the speeds are expressed, as we have established, from the square roots of the heights through which they have fallen, the speed will be as 15625 or as great as 125, in which time a body in one second can complete a distance of 31250 scruples. [For which the constant of proportionality can now be evaluated : thus a distance of 15625 corresponds to a speed of 125 in the square root proportionality, for which the same distance of 15625 scruples corresponds to an actual speed of 31250 scruples/sec. = 250 × 125; or the true speed = 250 × v for the height v fallen. Thus, the constant of proportionality is 250.]

Corollary 4

It is therefore easy to assign the distance that will be traversed in a time of one second with the speed expressed by v . Indeed it happens that the distances described in the same time are in the same ratio as the speeds, thus 125 is to v thus as 31250 scruples is to 250 v . For which by a factor 250 v expresses the distance in scruples traveled in one second, if indeed for the height v it may be shown by such a proportionality that the distance can be completed with a speed v in a second, and which is clearly is equal to the motion.

Example 1

The fall of a body from a height of 1000 ped., will be v = 1000000 in scruples, whereby for this descent it will acquire as much speed as in one second it travels a distance 250000 scruples, i. e. it will be able to complete 250 feet in one second.

Corollary 5

Reciprocally, if the speed is expressed by the distance that is traversed in one second, as we did in the beginning, then this can hence be reduced to our method of taking square roots for the corresponding heights. Indeed if that distance is a scrup., and the corresponding height for this speed is v scrup., then it follows that 250 v = a and ..

Example 2

Let the body have such a speed that in a second it is able to traverse a distance of 1000 feet or 1000000 scrup., then the height that corresponds to this speed will be 62500 0001000000000= scrup. or 1600 feet.

Scholium

Therefore the way in which each speed is to be expressed in turn is clear, and how it may be required that the one can be reduced to the other. For initially we were expressing the speeds by the distances travelled in a second, or in some other interval of time. Afterwards, truly it was seen that the speeds were shown to be in agreement with the corresponding heights. Now truly we show how each way can be adapted to measure the speed.