The Motion of Bodies Resisted in the Duplicate Ratio of Velocity on Superphysics

Projecticle

Mon, 01 Jan 0001 00:00:00 +0000

Proposition 6 Theorem 5

Homogeneous and equal spherical bodies, opposed by resistances that are in the duplicate ratio of the velocities, and moving on by their innate force only, will, in times which are reciprocally as the velocities at the beginning, describe equal spaces, and lose parts of their velocities proportional to the wholes.

Projecticle

Mon, 01 Jan 0001 00:00:00 +0000

If spherical bodies are resisted in the duplicate ratio of their velocities, in times which are as the first motions directly, and the first resistances inversely, they will lose parts of their motions proportional to the wholes, and will describe spaces proportional to those times and the first velocities conjunctly.

Projecticle

Mon, 01 Jan 0001 00:00:00 +0000

Lemma 2

The moment of any genitum is equal to the moments of each of the generatinrr sides drawn into the indices of the powers of those sides, and into their co-efficients continually.

Projecticle

Mon, 01 Jan 0001 00:00:00 +0000

PROPOSITION 8 THEOREM 6

If a body in an uniform medium, being uniformly acted upon by the force of gravity, ascends or descends in a right line; and the whole space described be distinguished into equal parts, and in the beginning of each of the parts (by adding or subducting the resisting force of the medium to or from the force of gravity, when the body ascends or descends] you collect the absolute forces; I say, that those absolute forces are in a geometrical progression.

Projecticle

Mon, 01 Jan 0001 00:00:00 +0000

Proposition 9 Theorem 7

Supposing what is above demonstrated, I say, that if the tangents of the angles of the sector of a circle, and of an hyperbola, be taken proportional to the velocities, the radius being of a fit magnitude, all the time of the ascent to the highest place will be as the sector of the circle, and all the time of descending from the highest place as the sector of the hyperbola.

Projecticle

Mon, 01 Jan 0001 00:00:00 +0000

Proposition 5 Theorem 3

If a body is resisted in the duplicate ratio of its velocity, and moves by its innate force only through a similar medium; and the times be taken in a geometrical progression, proceeding from less to greater terms: I say, that the velocities at the beginning of each of the times are in the same geometrical progression inversely; and that the spaces are equal, which are described in each of the times.