Page images
PDF
EPUB

quire the velocity with which the projectile was thrown. And when the projection is made with the elevation of 45° the height of the projection is one fourth of the greatest random.

If nothing be required of the engineer but to hit the object, it is better to use the constant angle of 45° and let the randoms be determined by the quantity of powder; because much powder will be saved by this means, and the engineer will be more certain of hitting his mark, than at any other elevation. An error of a degree in the elevation near 45 will not produce a sensible error in the distance which is proportional to the sines of double the angles of elevation. Now the sines of the angles near 90 do not differ much from the radius. The position is then evident.

The forces necessary to give a ball any degree of velocity, being as the square roots of the distances fallen through to acquire that velocity which will carry it over the proposed distance, must be as the square roots of these distances or horizontal ranges.

If it were therefore required to find what charge at an elevation of 45° would carry a ball over 3600 feet, from a cannon whose greatest random was 8000 feet, and the requisite charge of powder 28 pounds; this proportion will answer the question. As 80003600 :: 28:18 the charge required to carry the ball 3600 feet.

Still we are to remember, that this conclusion is mathematically true only upon the supposition that all the powder exerted its force in the bore of the piece. But we find, that some of it does not take fire, especially in high charges, until it be out of the mouth of the gun; which will occasion some variation, and an allowance must be made for it in practice. It is necessary also to

be remembered, that the heating of the cannon, after two or three shots, will lessen the cohesion of its particles, and make it more liable to burst; while the sudden drying of the powder will occasion more of it to take fire in the bore of the piece, and produce a greater force, either to overshoot the mark or split the cannon. On both of these accounts, the charges to be successively used, in a small space of time, should be gradually lessened.

If you wanted to find the impetus of a cannon, with a given charge of powder, let the cannon be shot off at the elevation of 45, and half the horizontal range is equal to the impetus, or distance from which a heavy body falls to acquire the velocity of the ball when it leaves the mouth of the cannon. This velocity is thus determined..

In order to find the velocity at parting from the cannon, we must remember, that the last acquired velocities in perpendicular descents are as the square roots of the spaces passed over, and that a descent through 16.15 feet will generate a uniform velocity of 32.3 feet per second. Therefore say, as 16: 32:: the square root of the impetus: the velocity of the ball at parting from the cannon; or so is the square root of half the greatest random, to the velocity. Or shorter, thus; since

32

8; eight times the square root of half the greatest

√16 range is the velocity of the ball, both when it leaves the cannon, and when it meets the horizon. Now this velocity multiplied by the weight of the ball, will give the momentum, or force, with which it strikes the object.

But in throwing bombs, which are not designed to do damage by their projectile force, we do not so much

regard their momentum, as the time of their flight, by which the length of their fuses is to be determined, that they may burst as soon as they fall to the horizon. Now, in order to compute the time of flight, we must remember that it is equal to the time of ascent and descent through the altitude of the projection: and that the times of descent through spaces, by the force of gravity, are as the square roots of the spaces; and also that a heavy body falls 16 feet in one second. Therefore say, as 16=4: 1 second of time: : the square root of the height of projection: seconds of time spent in the ascent; and this doubled is the time of flight in seconds when the height is measured in feet. Therefore half the square root of the height is the time of flight in seconds. But when the projection is made at the elevation of 45° the height of the projection is one fourth of the greatest random. Therefore at this elevation, one fourth of the square root of the greatest random is equal to the time of flight in seconds. And for other elevations, say, as sine 45 is to one fourth of the square root of the greatest range, so is the sine of the elevation, to the time of flight in seconds; the ranges being measured in feet.

The line of direction is always a tangent to the parabola in the place of the engine; because gravity, acting constantly upon the body as soon as it leaves the cannon, must bring it down towards the earth: in half a second causes it to descend 4 feet, in a whole second, 16 feet, and in two seconds, it will be found no less than 64 feet below the line of direction, when that is horizontal. Hence there is no such thing as a pointblank shot, as it is commonly called. Let the projectile motion of the ball be ever so rapid, gravity notwith

[graphic]

standing acts upon it as much as if there were no horizontal motion at all.

The line of elevation being a tangent to the curve, in the place of the engine, will bisect the angle between the zenith and the focus of the parabola, as is demonstrated by the writers on conic sections. It is also a known property of the parabola that the distance of the focus from any point in the curve is one fourth of the parameter of that point, or equal to the impetus of the engine; from whence it follows, that if a circle be described round the place of the engine with a radius equal to the impetus, it will pass through the focus of the parabola described with that impetus, and consequently through the foci of all the parabolas that can be described with that impetus, with any elevation. Hence also it follows, that when the elevation is 45°, the focus of that parabola will be in the horizon, and the foci of any two parabolas, that have the same horizontal random will be equally above and below the horizon, and the one perpendicularly above the other. It follows also, that if two circles be described round any two points of a parabola, with radii respectively equal to one fourth of their parameters; if they intersect each other it will be in two points, which will be the foci of two parabolas that pass through the centers of the circles; and these foci will be equally above and below the line that passes through the intersections of the parabolas or centers of the circles: but if the circles only touch one another, it will be in the focus of the only parabola that can pass through both the points, being described by the same impetus. Now should one of these points of the parabola be considered as the place of the engine, and the other an object to be struck, either above or below the horizon; the focus of the parabola will be in the line that

joins the engine and object, when the object is at the greatest distance that can be reached, on any plane either ascending or descending, with a given impetus. And the direction to hit the object will be the line that bisects the angle between the zenith and object. But if the object be at a less distance, the foci of the two parabolas that pass through the object will be equally above and below the line that connects the object and engine, and the directions to strike it will be equally above and below the line which bisects the angle between the zenith and the object.

MOTION OF BODIES ROUND A CENTER.

In our lecture on the motion of projectiles, we sup posed that the moving body was acted upon by an im. pressed force in a certain direction, called the projectile force, at the same time that gravity acted upon it in right lines parallel to each other; and consequently that the central force was placed at an infinite distance, and that it acted continually with the same degree of strength without any change arising from the projectile's approach to the center of motion, or recess from it. Both of these suppositions are necessary to cause the projectile to move in a parabola; though neither of them is strictly true: for gravity does not act in lines parallel to each other, but in lines directed to a center, and it increases in strength as the squares of the distances from the center decrease. But because of the great distance of the center of the earth from its surface, and the small distance to which we can throw any body, there is no sensible error committed in computing the projectile's motion ma parabola.

But when the distance of the central force bears a Considerable proportion to the distance which the pro

R

« PreviousContinue »