spaces as the squares of the times, or as the squares of the velocities.

For, since the force of gravity is uniform, and constantly the same, at all places near the earth's surface, or at nearly the same distance from the centre of the earth; and since this is the force by which bodies descend to the surface; they therefore descend by a force which acts constantly and equal. ly; consequently all the motions freely produced by gravity, are as above specified, by that proposition, &c.

SCHOLIUM.

154. Now it has been found, by numberless experiments, that gravity is a force of such a nature, that all bodies, whether light or heavy, fall vertically through equal spaces in the same time, abstracting from the resistance of the air; as lead or gold and a feather, which in an exhausted receiver fall from the top to the bottom in the same time. It is also found that the velocities acquired by descending, are in the exact proportion of the times of descent: and further, that the spaces descended are proportional to the squares of the times, and therefore to the squares of the velocities. Hence then it follows, that the weights or gravities, of bodies near the surface of the earth, are proportional to the quantities of matter contained in them; and that the spaces, times, and velocities, generated by gravity, have the relations contained in the three general proportions before laid down. Further, as it is found, by accurate experiments, that a body in the latitude of London, falls nearly 16, feet in the first second of time, and consequently that at the end of that time it has acquir. ed a velocity double, or of 321 feet by corol. 1, art. 132; therefore, ifg denote 16 feet, the space fallen through in one second of time, or g the velocity generated in that time; then, because the velocities are directly proportional to the times, and the spaces to the squares of the times; therefore it will be,

as 1"t": gt gt =v the velocity,
and 12:12::g: g
=s the space.

2

So that, for the descents of gravity, we have these general equations, namely,

Hence, because the times are as the velocities, and the spaces as the squares of either, therefore,

16, 25, &c.

and the spaces as their squares 1, 4, 9,

and the space for each time as 1, 3, 5, 7, 9, &c. namely, as the series of the odd numbers, which are the differences of the squares denoting the whole spaces. So that if the first series of natural numbers be seconds of time, namely, the times in seconds, 1", 2", 3", 4", &c. the velocities in feet will be 321, 641, 96, 128, the spaces in the whole times 16, 64, 144, 257, &c. and the space for each second 16, 481, 80, 112, &c. of which spaces the common difference is 32 feet, the natural and obvious measure of g, the force of gravity.

AN

a

d

с

B

&c.

155. These relations, of the times, ve. locities, and spaces, may be represented by certain lines and geometrical figures. Thus, if the line AB denote the time of any body's descent, and BC, at right angles to it, the velocity gained at the end of that time; by joining AC, and dividing the time AB into any number of parts at the points a, b, c ; then shall ad, be, cf, parallel to BC, be the velocities at the points of time, a, b, c, or at the ends of the times, aa, ab, Ac; because these latter lines, by similar triangles, are proportional to the former ad, be, cf, and the times are proportional to the velocities. Also, the area of the triangle ABC will represent the space descended by the force of gravity in the time AB, in which it generates the velocity BC; be cause that area is equal to AB X BC, and the space descended is s tv, or half the product of the time and the last velocity. And, for the same reason, the less triangles sad, Abe, acf, will represent the several spaces described in the corresponding times a, b, c, and velocities ad, be, cf; those triangles or spaces being also as the squares of their like sides aa, ab, Ac, which represent the times, or of ad, be, ef, which represent the velocities.

1=

156. But as areas are rather unnatural representations of the spaces passed over by a body in motion, which are lines, the relations may better be represented by the abscisses

Pabcd

e

and ordinates of a parabola. Thus, if po be a parabola, PR its axis, and RQ its ordinate; and ra, rb, rc, &c. parallel to RQ, represent the times from the beginning, or the velocities, then ae, bf, cg, &c. parallel to the axis PR, will represent the 7 spaces described by a falling body in those times; for, in a parabola, the abscisses ph, ri, rk, &c. or ae, bf, cg, &c. which are the spaces described, are as the squares of the ordinates he, if, kg, &c. or ra, rb, rc, &c. which represent the times or velocities.

R

157. And because the laws for the destruction of motion, are the same as those for the generation of it, by equal forces, but acting in a contrary direction; therefore,

1st, A body thrown directly upward, with any velocity, will lose equal velocities in equal times.

2d, If a body be projected upward, with the velocity it acquired in any time by descending freely, it will lose all its velocity in an equal time, and will ascend just to the same height from which it fell, and will describe equal spaces in equal times, in rising and falling, but in an inverse order; and it will have equal velocities at any one and the same point of the line described, both in ascending and descending.

3d, If bodies be projected upward, with any velocities, the height ascended to, will be as the squares of those velocities, or as the squares of the times of ascending, till they lose all their velocities.

158. In solving problems, where a body, instead of being permitted to fall freely, is projected vertically upwards or downwards with a given velocity, it will assist the comprehension of what takes place, to ascertain what results from the original projection, and what from the force of gravity. Thus, if a body be projected with a velocity v, it will, in the time t, described the space tv (art. 129) apart from the opera. tion of gravity or any other force. Blending this with the preceding expression for the space described by a falling body, we have

s = tv = gt2,

in which the lower sign must be employed when the projec. tion is vertically downwards, the upper when the projection is vertically upwards.

1. Find the space descended vertically by a body in 7 seconds of time, and the velocity acquired?

Ans. 788, space; 225, velocity,

2. Required the time of generating a velocity of 100 feet. per second, and the whole space descended.

Ans. 3", time; 155, f. space.

3. Find the time of descending 400 feet, and the velocity at the end of that time.

Ans. 4", time; 1603, velocity. 4. If a body fall freely for 5", how far will it descend during the last second of its motion ?

5. If an arrow be propelled vertically upwards from a bow with a velocity of 96 feet per second, how high will it rise, and how long will it be before it returns again to the ground?

6. If a ball be projected vertically downwards with a velocity of 100 feet per second, how far will it have descended in three seconds?

7. If a ball be projected upwards with a velocity of 100 feet per second, how far will it have arisen in three seconds?

8. If a ball be projected vertically upwards with a velocity of 44 feet per second, will it be above or below the point of projection in four seconds, the force of gravity tending all the time to draw it downwards ?

9. A drop of rain falls through 176 feet in the last second; how high is the cloud from which it descended?

10. A body falling freely was observed to pass through half its descent in the last second; how far did it fall, and how long was it in falling?

11. Two weights, one of 5lbs. the other of 3lbs. hang freely over a pulley: after motion is allowed to commence how far will the larger weight descend, or the smaller arise, in four seconds?

12. Two equal weights are balanced over a pulley. A pound weight being added to one of them, and motion in consequence taking place, the preponderating weight descended through 16 feet in four seconds. Required the measure of the two equal weights?

158. PROP. If a body be projected in free spice, either pa. ral'el to the horizon, or in an oblique direction, by the force of gunpowder, or any other impulse; it will by this motion, in conjunction with the action of gravity, describe the curve line of a parabola.

Let the body be projected from the point a, in the direction AD, with any uniform velocity: then, in any equal portions of time, it would, by art. 129, describe the equal spaces AB, BC, CD, &c. in the line AD, if it were not drawn continually down below that line by the action of gravity. Draw BE, CF, DG, &c. in the direction of gravity, or perpen. dicular to the horizon, and equal to the spaces through which the body would descend by its gravity in the same time in which it would uniformly pass over the correspond, ing spaces AB, AC, AD, &c. by the projectile motion. Then, since by these two motions the body is carried over the space AB, in the same time as over the space BE, and the space a in the same time as the space cr, and the space AD in the same time as the space DG, &c. ; therefore, by the composition of motions, at the end of those times, the body will be found respectively in the points E, F, G, &c.; and con, sequently the real path of the projectile will be the curve line AEFG, &c. But the spaces AB, AC, AD, &c. described by uniform motion, are as the times of description; and the spaces BE, CF, DG, &c. described in the same times by the accelerating force of gravity, are as the squares of the times; consequently the perpendicular descents are as the squares of the spaces in AD, that is BE, CF, DG, &c. are respectively proportional to ab2, ac2, ad3, &c. ; which is the property of the parabola by theor. 8, Con. Sect. Therefore the path of the projectile is the parabolic line AEFG, &c. to which AD is a tangent at the point a.

159. Corol. 1. The horizontal velocity of a projectile, is always the same constant quantity, in every point of the curve because the horizontal motion is in a constant ratio to the motion in AD, which is the uniform projectile motion. And the projectile velocity is in proportion to the constant horizontal velocity, as radius to the cosine of the angle DAH, or angle of elevation or depression of the piece above or below the horizontal line AH.