PROJECTILES

, the science of the motion, velocity, flight, range, &c, of a projectile put into violent motion by some external cause, as the force of gunpowder, &c. This is the foundation of gunnery, under which article may be found all that relates peculiarly to that branch.

All bodies, being indifferent as to motion or rest, will necessari<*>y continue the state they are put into, except so far as they are hindered, and forced to change it by some new cause. Hence, a Projectile, put in motion, must continue eternally to move on in the same right line, and with the same uniform or constant velocity, were it to meet with no resistance from the medium, nor had any force of gravity to encounter.

In the first case, the theory of Projectiles would be very simple indeed; for there would be nothing more to do, than to compute the space passed over in a given time by a given constant velocity; or either of these, from the other two being given.

But by the constant action of gravity, the Projectile is continually deflected more and more from its right-lined course, and that with an accelerated velocity; which, being combined with its Projectile impulse, causes the body to move in a curvilineal path, with a variable motion, which path is the curve of a parabola, as will be proved below; and the determination of the range, time of flight, angle of projection, and variable velocity, constitutes what is usually meant by the doctrine of Projectiles, in the common acceptation of the word.

What is said above however, is to be understood of Projectiles moving in a non-resisting medium; for when the resistance of the air is also considered, which is enormously great, and which very much impedes the first Projectile velocity, the path deviates greatly from the parabola, and the determination of the circumstances of its motion becomes one of the most complex and difficult problems in nature.

In the first place therefore it will be proper to consider the common doctrine of Projectiles, or that on the parabolic theory, or as depending only on the nature of gravity and the Projectile motion, as abstracted from the resistance of the medium.

Little more than 200 years ago, philosophers took the line described by a body projected horizontally, such as a bullet out of a cannon, while the force of the powder greatly exceeded the weight of the bullet, to be a right line, after which they allowed it became a curve. Nicholas Tartaglia was the first who perceived the mistake, maintaining that the path of the bullet was a curved line through the whole of its extent. But it was Galileo who first determined what particular curve it is that a Projectile describes; shewing that the path of a bullet projected horizontally from an eminence, was a parabola; the vertex of which is the point where the bullet quits the cannon. And the same is proved generally, in the 2d section following, when the projection is made in any direction whatever, viz, that the| curve is always a parabola, supposing the body moves in a non-resisting medium.

I. If a heavy body be projected perpendicularly, it will continue to ascend or descend perpendicularly; because both the projecting and the gravitating force are found in the same line of direction.

II. If a body be projected in free space, either parallel to the horizon, or in any oblique direction; it will, by this motion, in conjunction with the action of gravity, describe the curve line of a parabola.

For let the body be projected from A, in the direction AD, with any uniform velocity; then in any equal portions of time it would, by that impulse alone, 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 perpendicular to the horizon; and take BE, CF, DG, &c, equal to the spaces through which the body would descend by its gravity in the same times in which it would uniformly pass over the spaces AB, AC, AD, &c, by the Projectile motion. Then, since by these motions, the body is carried over the space AB in the same time as the space BE, and the space AC in the same time as the space CF, 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 consequently the real path of the Projectile will be the curve line AEFG &c. But the spaces AB, AC, AD, &c, being 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, AD², &c, which is the same as the property of the parabola. Therefore the path of the Projectile is the parabolic line AEFG &c, to which AD is a tangent at the point A.

Hence, 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 con- stant ratio to the motion in AD, which is the uniform Projectile motion; viz, the constant horizontal velocity being to the Projectile 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.

2. The velocity of the Projectile in the direction of the curve, or of its tangent, at any point A, is as the secant of its angle BAI of direction above the horizon. For the motion in the horizontal direction AI being constant, and AI being to AB as radius to the secant of the angle A; therefore the motion at A, in AB, is as the secant of the angle A.

3. The velocity in the direction DG of gravity, or perpendicular to the horizon, at any point G of the curve, is to the first uniform Projectile velocity at A, as 2GD to AD. For the times of describing AD and DG being equal, and the velocity acquired by freely descending through DG being such as would carry the body uniformly over twice DG in an equal time, and the spaces described with uniform motions being as the velocities, it follows that the space AD is to the space 2DG, as the Projectile velocity at A is to the perpendicular velocity at G.

III. The velocity in the direction of the curve, at any point of it, as A, is equal to that which is generated by gravity in freely descending through a space which is equal to one-fourth of the parameter of the diameter to the parabola at that point.

Let PA or AB be the height due to the velocity of the Projectile at any point A, in the direction of the curve or tangent AC, or the velocity acquired by falling through that height; and complete the parallelogram ACDB. Then is CD = AB or AP the height due to the velocity in the curve at A; and CD is also the height due to the perpendicular velocity at D, which will therefore be equal to the former: but, by the last corollary, the velocity at A is to the perpendicular velocity at D, as AC to 2CD; and as these velocities are equal, therefore AC or BD is equal to 2CD or 2AB; and hence AB or AP is equal to 1/2BD or 1/4 of the parameter of the diameter AB by the nature of the parabola.

Hence, 1. If through the point P, the line PL be drawn perpendicular to AP; then the velocity in the curve at every point, will be equal to the velocity acquired by falling through the perpendicular distance| of the point from the said line PL; that is, a body falling freely through PA, acquires the velocity in the curve at A, EF, " at F, KD, " at D, LH, " at H. The reason of which is, that the line PL is what is called the Directrix of the parabola, the property of which is, that the perpendicular to it, from every point of the curve, is equal to one-fourth of the parameter of the diameter at that point, viz, PA = 1/4 the parameter of the diameter at A, EF = " at F, KD = " at D, LH = " at H.

2. If a body, after falling through the height PA, which is equal to AB, and when it arrives at A if its course be changed, by reflection from a firm plane AI, or otherwise, into any direction AC, without altering the velocity; and if AC be taken equal to 2AP or 2AB, and the parallelogram be completed; the body will describe the parabola passing through the point D.

3. Because AC = 2AB or 2CD or 2AP, therefore AC² = 2AP . 2CD or AP . 4CD; and because all the perpendiculars EF, CD, GH are as AE², AC², AG²; therefore also AP . 4EF = AE², and AP . 4GH = AG², &c; and because the rectangle of the extremes is equal to the rectangle of the means, of four proportionals, therefore it is always, , and , and , and so on.

IV. Having given the Direction of a Projectile, and the Impetus or Altitude due to the sirst velocity; to determine the Greatest Height to which it will rise, and the Random or Horizontal Range.

Let AP be the height due to the Projectile velocity at A, or the height which a body must fall to acquire the same velocity as the projectile has in the curve at A; also AG the direction, and AH the horizon. Upon AG let fall the perpendicular PQ, and on AP the perpendicular QR; so shall AR be equal to the greatest altitude CV, and 4RQ equal to the horizontal range AH. Or, having drawn PQ perpendicular to AG, take AG = 4AQ, and draw GH perpendicular to AH; then AH is the range.

For by the last cor. ... , and by sim.triangles, ... , or ; therefore AG = 4AQ; and, by similar triangles, AH = 4RQ.

Also, if V be the vertex of the parabola, then AB or 1/2AG = 2AQ, or AQ = QB; consequently AR = BV which is = CV by the nature of the parabola.

Hence, 1. Because the angle Q is a right angle, which is the angle in a semicircle, therefore if upon AP as a diameter a semicircle be described, it will pass through the point Q.

2. If the Horizontal Range and the Projectile Velocity be given, the Direction of the piece so as to hit the object H will be thus easily found: Take AD = 1/4AH, and draw DQ perpendicular to AH, meeting the semicircle described on the diameter AP in Q and q; then either AQ or Aq will be the direction of the piece. And hence it appears, that there are two directions AB and Ab which, with the same Projectile velocity, give the very same horizontal range AH; and these two directions make equal angles qAD and QAP with AH and AP, because the arc PQ is equal to the arc Aq.

3. Or if the Range AH and Direction AB be given; to find the Altitude and Velocity or Impetus: Take AD = 1/4AH, and erect the perpendicular DQ meeting AB in Q; so shall DQ be equal to the greatest altitude CV. Also erect AP perpendicular to AH, and QP to AQ; so shall AP be the height due to the velocity.

4. When the body is projected with the same velocity, but in different directions; the horizontal ranges AH will be as the sines of double the angles of elevation. Or, which is the same thing, as the rectangle of the sine and cosine of elevation. For AD or RQ, which is 1/4AH, is the sine of the arc AQ, which measures double the angle QAD of elevation.

And when the direction is the same, but the velocities different, the horizontal ranges are as the square of the velocities, or as the height AP which is as the square of the velocity; for the sine AD or RQ, or 1/4AH, is as the radius, or as the diameter AP

Therefore, when both are different, the ranges are in the compound ratio of the squares of the velocities, and the sines of double the angles of elevation.

5. The greatest range is when the angle of elevation is half a right angle, or 45°. For the double of 45 is 90°, which has the greatest sine. Or the radius OS, which is 1/4 of the range, is the greatest sine.

And hence the greatest range, or that at an elevation of 45°, is just double the altitude AP which is due to| the velocity. Or equal to 4VC. And consequently, in that case, C is the focus of the parabola, and AH its parameter.

And the ranges are equal at angles equally above and below 45°.

6. When the elevation is 15°, the double of which, or 30°, having its sine equal to half the radius, consequently its range will be equal to AP, or half the greatest range at the elevation of 45°; that is, the range at 15° is equal to the impetus or height due to the projectile velocity.

7. The greatest altitude CV, being equal to AR, is as the versed sine of double the angle of elevation, and also as AP or the square of the velocity. Or as the square of the sine of elevation, and the square of the velocity; for the square of the sine is as the versed sine of the double angle.

8. The time of flight of the projectile, which is equal to the time of a body falling freely through GH or 4CV, 4 times the altitude, is therefore as the square root of the altitude, or as the projectile velocity and sine of the elevation.

9. And hence may be deduced the following set of theorems, for finding all the circumstances relating to projectiles on horizontal planes, having any two of them given. Thus, let s, c, t = sine, cosine, and tang. of elevation, S, v = sine and vers. of double the elevation, R the horizontal range, T the time of flight, V the projectile velocity, H the greatest height of the projectile, g = 16 1/12 feet, and a = the impetus or the altitude due to the velocity V. Then, .

And from any of these, the angle of direction may be found.

V. To determine the Range on an oblique plane; having given the Impetus or the Velocity, and the Angle of Direction.

Let AE be the oblique plane, at a given angle above or below the horizontal plane AH; AG the direction of the piece; and AP the altitude due to the projectile velocity at A.

By the last prop. find the horizontal range AH to the given velocity and direction; draw HE perpendicular to AH meeting the oblique plane in E; draw EF parallel to the direction AG, and FI parallel to HE; so shall the projectile pass through I, and the range on the oblique plane will be AI. This is evident from prob. 17 of the Parabola in my treatise on Conic Sections, where it is proved, that if AH, AI be any two lines terminated at the curve, and IF, HE be parallel to the axis; then is EF parallel to the tangent AG.

Hence, 1. If AO be drawn perpendicular to the plane AI, and AP be bisected by the perpendicular STO; then with the centre O describing a circle through A and P, the same will also pass through q, because the angle GAI, formed by the tangent AG and AI, is equal to the angle APq, which will therefore stand upon the same arc Aq.

2. If there be given the Range and Velocity, or the Impetus, the Direction will then be easily found thus: Take Ak = 1/4AI, draw kq perpendicular to AH, meeting the circle described with the radius AO in two points q and q; then Aq or Aq will be the direction of the piece. And hence it appears that there are two directions, which, with the same impetus, give the very same range AI, on the oblique plane. And these two directions make equal angles with AI and AP, the plane and the perpendicular, because the arc Pq = the arc Aq. They also make equal angles with a line drawn from A through S, because the arc Sq = the arc Sq.

3. Or, if there be given the Range AI, and the Direction Aq; to find the Velocity or Impetus. Take Ak = 1/4AI, and erect kq perpendicular to AH meeting the line of direction in q; then draw qP making the angle AqP = the angle Akq; so shall AP be the impetus, or altitude due to the projectile velocity.

4. The range on an oblique plane, with a given elevation, is directly as the rectangle of the cosine of the direction of the piece above the horizon and the sine of the direction above the oblique plane, and reciprocally as the square of the cosine of the angle of the plano above or below the horizon.|

For put s = sin. [angle] qAI or APq, c = cos. [angle] qAH or sin. PAq, C = cos. [angle] IAH or sin. Akd or Akq or AqP. Then, in the tri. APq, ... , and in the tri. Akq, ... , therefore by compos. ... .

So that the oblique range .

Hence the range is the greatest when Ak is the greatest, that is when kq touches the circle in the middle point S, and then the line of direction passes through S, and bisects the angle formed by the oblique plane and the vertex. Also the ranges are equal at equal angles above and below this direction for the maximum.

5. The greatest height tv or kq of the projectile, above the plane, is equal to . And therefore it is as the impetus and square of the sine of direction above the plane directly, and square of the cosine of the plane's inclination reciprocally. For C (sin. AqP) : s (sin. APq) :: AP : Aq, and C (sin. Akq) : s (sin. kAq) :: Aq : kq, thereforeby comp. C² : s² :: AP : kq.

6. The time of flight in the curve AvI is = , where g = 16 1/12 feet. And therefore it is as the velocity and sine of direction above the plane directly, and cosine of the plane's inclination reciprocally. For the time of describing the curve, is equal to the time of falling freely through GI or 4kq or . Therefore, the time being as the square root of the distance, the time of flight.

7. From the foregoing corollaries may be collected the following set of theorems, relating to projects made on any given inclined planes, either above or b<*>low the horizontal plane. In which the letters denote as before, namely, c = cos. of direction above the horizon, C = cos. of inclination of the plane, s = sin. of direction above the plane, R the range on the oblique plane, T the time of flight, V the projectile velocity, H the greatest height above the plane, a the impetus, or alt. due to the velocity V, g = 16 1/12 feet. Then . And from any of these, the angle of direction may be found.

For a long time after Galileo, philosophers seemed to be satisfied with the parabolic theory of Projectiles, deeming the effect of the air's resistance on the path as of no consequence. In process of time, however, <*>as the true philosophy began to dawn, they began to suspect that the resistance of the medium might have some effect upon the Projectile curve, and they set themselves to consider this subject with some attention.

Huygens, supposing that the resistance of the air was proportional to the velocity of the moving body, concluded that the line described by it would be a kind of logarithmic curve.

But Newton, having clearly proved, that the resistance to the body is not proportional to the velocity itself, but to the square of it, shews, in his Principia, that the line a Projectile describes, approaches nearer to an hyperbola than a parabola. Schol. prop. 10, lib. 2. Thus, if AGK be a curve of the hyperbolic kind, one of whose asymptotes is NX, perpendicular to the horizon AK, and the other IX inclined to the same, where VG is reciprocally as DNⁿ, whose index is n: this curve will nearer represent the path of a Projectile thrown in the direction AH in the air, than a parabola. Newton indeed says, that these hyperbolas are not accurately the curves that a Projectile makes in the air; for the true ones are curves which about the vertex are more distant from the asymptotes, and in the parts remote from the axis approach nearer to the asymptotes than these hyperbolas; but that in practice these hyperbolas may be used instead of those more compounded ones. And if a body be projected from A, in the right line AH, and AI be drawn parallel to the asymptote NX, and GT a tangent to the curve at the vertex: Then the density of the medium in A will be reciprocally as the tangent AH, and the body's velocity will be as , and the resistance of the medium will be to gravity, as AH to .

M. John Bernoulli constructed this curve by means of the quadrature of some transcendental curves, at the| request of Dr. Keil, who proposed this problem to him in 1718. It was also resolved by Dr. Taylor; and another solution of it may be found in Hermann's Phoronomia.

The commentators Le Sieur and Jacquier say, that the description of the curve in which a Projectile moves, is so very perplexed, that it can scarcely be expected any deduction should be made from it, either to philosophical or mechanical purposes: vol. 2. pa. 118.

Dan. Bernoulli too proved, that the resistance of the air has a very great esfect on swift motions, such as those of cannon shot. He concludes from experiment, that a ball which ascended only 7819 feet in the air, would have ascended 58750 feet in vacuo, being near eight times as high. Comment. Acad. Petr. tom. 2.

M. Euler has farther investigated the nature of this curve, and directed the calculation and use of a number of tables for the solution of all cases that occur in gunncry, which may be accomplished with nearly as much expedition as by the common parabolic principles. Memoirs of the Academy of Berlin, for the year 1753.

But how rash and erroneous the old opinion of the inconsiderable resistance of the air is, will easily appear from the experiments of Mr. Robins, who has shewn that, in some cases, this resistance to a cannon ball, amounts to more than 20 times the weight of the ball; and I myself, having prosecuted this subject far beyond any former example, have sometimes found this resistance amount to near 100 times the weight of the ball, viz, when it moved with a velocity of 2000 feet per second, which is a rate of almost 23 miles in a minute. What errors then may not be expected from an hypothesis which neglects this force, as inconsiderable! Indeed it is easy to shew, that the path of such Projectiles is neither a parabola nor nearly a parabola. For, by that theory, if the ball, in the instance last mentioned, flew in the curve of a parabola, its horizontal range, at 45° elevation, will be found to be almost 24 miles; whereas it often happens that the ball, with such a velocity, ranges far short of even one mile.

Indeed the falseness of this hypothesis almost appears at sight, even in Projectiles slow enough to have their motion traced by the eye; for they are seen to descend through a curve manifestly shorter and more inclined to the horizon than that in which they ascended, and the highest point of their flight, or the vertex of the curve, is much nearer to the place where they fall on the ground, than to that from whence they were at first discharged. These things cannot for a moment be doubted of by any one, who in a proper situation views the flight of stones, arrows, or fhells, thrown to any confiderable distance.

Mr. Robins has not only detected the errors of the parabolic theory of gunnery, which takes no account of the resistance of the air, but shews how to compute the real range of resisted bodies. But for the method which he proposes, and the tables he has computed for this purpose, see his Tracts of Gunnery, pa. 183, &c, vol. 1; and also Euler's Commentary on the same, translated by Mr. Hugh Brown, in 1777.

There is an odd circumstance which often takes place in the motion of bodies projected with considerable force, which shews the great complication and difficulty of this subject; namely, that bullets in their flight are not only depressed beneath their original direction by the action of gravity, but are also frequently driven to the right or left of that direction by the action of some other force.

Now if it were true that bullets varied their direction by the action of gravity only, then it ought to happen that the errors in their flight to the right or left of the mark they were aimed at, fhould increase in the proportion of the distance of the mark from the piece only. But this is contrary to all experience; the same piece which will carry its bullet within an inch of the intended mark, at 10 yards distance, cannot be relied on to 10 inches in 100 yards, much less to 30 in 300 yards.

And this inequality can only arise from the track of the bullet being incurvated sideways as well as downwards; for by this means the distance between the incurvated line and the line of direction, will increase in a much greater ratio than that of the distance; these lines coinciding at the mouth of the piece, and afterwards separating in the manner of a curve from its tangent, if the mouth of the piece be considered as the point of contact.

This is put beyond a doubt from the experiments made by Mr. Robins; who found alfo that the direction of the shot in the perpendicular line was not less uncertain, falling sometimes 200 yards short of what it did at other times, although there was no visible cause of difference in making the experiment. And I myself have often experienced a difference of one-fifth or one-sixth of the whole range, both in the deflection to the right or left, and also in the extent of the range, of cannon shot.

If it be asked, what can be the cause of a motion so different from what has been hitherto supposed? It may be answered, that the deflection in question must be owing to some power acting obliquely to the progressive motion of the body, which power can be no other than the resistance of the air. And this resistance may perhaps act obliquely to the progressive motion of the body, from inequalities in the resisted surface; but its general cause is doubtless a whirling motion acquired by the bullet about an axis, by its friction against the sides of the piece; for by this motion of rotation, combined with the progressive motion, each part of the ball's surface will strike the air in a direction very different from what it would do if there was no such whirl; and the obliquity of the action of the air, arising from this cause, will be greater, according as the rotatory motion of the bullet is greater in proportion to its progressive motion. Tracts, vol. 1, p. 149, &c.

M. Euler, on the contrary, attributes this deflection of the ball to its sigure, and very little to its rotation: for if the ball was perfectly round, though its centre of gravity did not coincide, the deflection from the axis of the cylinder, or line of direction sideways, would be very inconsiderable. But when it is not round, it will| generally go to the right or left of its direction, and so much the more, as its range is greater. From his reasoning on this subject he infers, that cannon shot, which are made of iron, and rounder and less susceptible of a change of sigure in passing along the cylinder than those of lead, are more certain than musket shot. True Principles of Gunnery investigated, 1777, p. 304, &c.