11. Given the Range at One Elevation; to find the Range at Another Elevation.

RULE. As the sine of double the first elevation, is to its range; so is the sine of double another elevation, to its range.

III. Given the Range for one Charge; to find the Range for Another Charge, or the Charge for Another Range.

RULE. The ranges have the same proportion as the charges; that is, as one range is to its charge, so is any other range to its charge: the elevation of the piece being the same in both cases.

192. EXAMPLE 1. If a ball of 1lb. acquire a velocity of 1600 feet per second, when fired with 8 ounces of powder; it is required to find with what velocity each of the several kinds of shells will be discharged by the full charges of pow. der, viz.

EXAM. 2. If a shell be found to range 1000 yards when discharged at an elevation of 45°; how far will it range when the elevation is 30° 16′, the charge of powder being the same? Ans. 2612 feet, or 871 yards.

EXAM. 3. The range of a shell, at 45° elevation, being found to be 3750 feet; at what elevation must the piece be set, to strike an object at the distance of 2810 feet, with the same charge of powder? Ans. at 24° 16′, or at 65° 44'.

EXAM. 4. With what impetus, velocity, and charge of powder, must a 13-inch shell be fired, at an elevation of 32° 12', to strike an object at the distance of 3250 feet?

Ans. impetus 1802, veloc. 340, change 4lb. 7 oz. EXAM. 5. A shell being found to range 3500 feet, when

3c

Sir Tho. Blomfield and Dr. Hutton made their experiments. Putting B for the weight of the ball, and c for that of the charge, v = 16001is now found a good approximative theorem for the initial velocity.

discharged at an elevation of 25° 12'; how far then will it fange at an elevation of 36° 15' with the same charge of powder? Ans. 4332 feet. EXAM. 6. If, with a charge of 9lb. of powder, a shell range 4000 feet; what charge will suffice to throw it 3000 feet, the elevation being 45° in both cases?

Ans. 63lb. of powder. EXAM. 7. What will be the time of flight for any given range, at the elevation of 45°, or for the greatest range ?

Ans. the time in secs. is the sq. root of the range in feet. EXAM. 8. In what time will a shell range 3250 feet, at an elevation of 32°? Ans. 11 sec. nearly.

EXAM. 9. How far will a shot range on a plane which ascends 8° 15', and another which descends 8° 15′; the im. petus being 3000 feet, and the elevation of the piece 32° 30'?

Ans. 4244 feet on the ascent, and 6745 feet on the descent.

EXAM. 10. How much powder will throw a 13-inch shell 4244 feet on an inclined plane, which ascends 8° 15', the elevation of the mortar being 32° 30′ ?

Ans. 7.3765lb. or 7lb. 6oz. EXAM. 11. At what elevation must a 13-inch mortar be pointed, to range 6745 feet, on a plane which descends 8° 15'; the charge 73lb. of powder? Ans. 32° 41'4.

EXAM. 12. In what time will a 13-inch shell strike a plane which rises 8° 30', when elevated 45°, and discharged with an impetus of 2304 feet? Ans. 14 seconds.

THE DESCENT OF BODIES ON INCLINED PLANES AND CURVE SURFACES.

PENDULUMS.

THE MOTION OF

193. PROP. IF a weight w be sustained on an inclined plane AB, by a power r, acting in a direction wr, parallel to the plane. Then

The weight of the body, w,
The sustaining power P, and
The pressure on the plane, p,
are respectively as

The length AB,

The height BC, and
The base AC,
of the plane.

For, draw CD perpendicular to the plane. Now here are three forces, keeping one another in equilibrio; namely, the weight, or force of gravity, acting perpendicular to ac, or parallel to BC; the power acting parallel to DB; and the pressure

A

P

perpendicular to AB, or parallel to DC: but when three forces keep one another in equilibrio, they are proportional to the sides of the triangle CBD, made by lines in the direction of those forces, by art. 30; therefore those forces are to one another as Bc, bd, cd. But the two triangles ABC, CBD, are equiangular, and have their like sides proportional; there. fore the three BC, BD, CD, are to one another respectively as the three AB, BC, AC; which therefore are as the three forces W, P, P.

Corol. 1. Hence the weight w, power P, and pressure p, are respectively as radius, and cosine, of

sine

the plane's elevation BAC above the horizon.

For, since the sides of triangles are as the sines of their opposite angles, therefore the three AB, BC, AC,

are respectively as

or as

sin. c, sin. A, sin. B,

radius, sine, cosine,
of the angle A of elevation.

Corol. 2. The power or relative weight that urges a body

BC

w down the inclined plane, is = X w; or the force with

AB

which it descends, or endeavours to descend, is as the sine of the angle a of inclination.

Corol. 3. Hence, if there be two planes of the same height, and two bodies be laid on them which are proportional to the lengths of the planes; they will have an equal tendency to descend down the planes. And consequently they will mutu. ally sustain each other if they be connected by a string parallel to the planes. Corol. 4. In like manner, when the power P acts in any other direction whatever, wr; by draw. ing CDE perpendicular to the direction wr, the three forces in equilibrio, namely, the weight w, the power P, and the pressure on the plane, will still be respectively

W

B

as AC, CD, AD, drawn perpendicular to the direction of those forces.

194. PROP. The velocity acquired by a body descending freely down an inclined plane AB, is to the velocity acquired by a body falling perpendicularly, in the same time; as the height of the plane BC, is to its length AB.

B

Ꭰ

For the force of gravity, both perpendicularly and on the plane, is constant; and these two, by corol. 2, art. 193, are to each other as AB to BC. But, by art. 131, the velocities gene. rated by any constant forces, in the same time, are as those forces. Therefore the velocity down BA is to the velocity down BC, in the same time, as the force on BA to the force on BC: that is, as BC to Ba.

Corol. 1. Hence, as the motion down an inclined plane is produced by a constant force, it will be a motion uniformly accelerated; and therefore the laws before laid down for accelerated motions in general, hold good for motions on inclined planes; such, for instance, as the following: That the velocities are as the times of descending from rest; that the spaces descended are as the squares of the velocities, or squares of the times; and that if a body be thrown up an inclined plane, with the velocity it acquired in descending, it will lose all its motion, and ascend to the same height, in the same time, and will repass any point of the plane with the same velocity as it passed it in descending.

Corol. 2. Hence also, the space descended along an inclined plane, is to the space descended perpendicularly, in the same time, as the height of the plane CB, to its length AB, or as the sine of inclination to radius. For the spaces described by any forces, in the same time, are as the forces, or as the velocities.

Corol. 3. Consequently the velocities and spaces descended by bodies down different inclined planes, are as the sines of elevation of the planes.

Corol. 4. If CD be drawn perpendicular to AB; then, while a body falls freely through the perpendicular space BC, another body will, in the same time, descend down the part of the plane BD. For, by similar triangles, BC BD BA BC, that is, as the space descended, by corol. 2.

Or, in any right-angled triangle BDC, having its hypothenuse BC perpendicular to the horizon, a body will descend down any of its three sides, BD, BC, DC, in the same time. And therefore, if on the diameter BC a circle be described, the

time of descending down any chords BD, BE, BF, DC, FC, FC, &c. will be all equal, and each equal to the time of falling freely through the perpendicular diameter BC. Also the velocities acquired in descending down the chords BD, BE, BF, BC, are to one another as the lengths of those chords.

E

195. PROP. The time of descending down the inclined plane BA, is to the time of falting through the height of the plane BC, as the length BA is to the height BC.

Draw CD perpendicular to AB. Then the times of describing BD and BC are equal, by the last corol. Call that time 1, and the time of describ. ing BA call T.

Now, because the spaces described

A

B

by constant forces, are as the squares of the times; therefore 12: T2:: BD: EA.

But the three BD, BC, BA, are in continual proportion; therefore BD: BA :: BC2: BA2;

hence, by equality, : T:: BC: BA2,

or

t:T: BC: BA.

Corol. Hence the times of descending different planes, of the same height, are to one another as the lengths of the planes.

196. PROF. A body acquires the same velocity in descending down any inclined plane BA, as by falling perpendicular through the height of the plane BC.

For, the velocities generated by any constant forces, are in the compound ratio of the forces and times of acting. But if we put

F to denote the whole force of gravity in BC,

f the force on the plane AB,

t the time of describing BC, and

T the time of descending down AB; then by art. 193, F:ƒ :: BA: BC; and by art, 195, t:T:: BC: Ba; theref. by comp. Ft: fr:: 1 : 1.

B

E

That is, the compound ratio of the forces and times, or the ratio of the velocities, is a ratio of equality.

Corol. 1. Hence the velocities acquired, by bodies des