Advancements In The Quest To Solve Newton's Riddle

by Robert E. Wall, Jr. Lt. Col. USAF (Ret.) 270 Muscovy Trail Sumter, SC. 29150 rwrockywall@gmail.com 1-803-968-9631 (c) 1-803-469-4014 (h)

Feb 28, 2013 revisions Mar 1, 2015

Abstract

In the field of physics and in particular that of ballistics, a problem has been attributed to Sir Isaac Newton. He, supposedly, proposed a problem, some 350 years ago, to provide a closed form solution to the trajectories of objects either tossed or fired into the air considering the resistance of the atmosphere. This is a classic problem taught in all high school and college physics courses. However, the problem has been over simplified in order to obtain a solution. The equations of motion have never been solved when atmospheric drag has been properly taken into account.

While the equations of motion still have not been successfully integrated with respect to time, the author has made some valuable progress toward a solution by integrating the equations with respect to the flightpath angle. The equations are shown to be solvable in this manner. However, some solutions necessarily had to be left as unevaluated integrals. Special case scenarios are also shown with their closed form solutions. Finally, the paper provides ideas and recommendations for future research.

Introduction

Most basic physics and dynamics textbooks deal with the motion of projectiles thrown or fired into the atmosphere from the surface of the earth and solve the equations of motion based on the assumption that the atmospheric resistance, called drag, is negligible. (Ref 1)

This assumption may be fine for educational purposes but it may not be at all valid in the real world depending upon the speed of the projectile, the relative amount of drag, and the degree of accuracy desired.

Some streamlined objects have a relatively small amount of drag and it may be acceptable to ignore this. However, very slow objects traveling through the atmosphere demonstrate a resistance proportional to the first power of the velocity. While higher speed objects such as subsonic aircraft will experience a drag proportional to the second power of the velocity. Also, supersonic aircraft and projectiles will experience drag that is even more complex, varying in an almost unpredictable fashion near the speed of sound.

The types of mathematical solutions involved will, naturally, vary greatly depending upon the situations described above and the assumptions made by the mathematician. When the atmospheric drag is ignored, the equations are very simple and are easily solved using ordinary algebra. These solutions are well known and appear commonly in basic math and physics textbooks (Ref 1).

When drag is considered for slow moving objects (less than a few tens of meters per second, i.e. the first power drag case), the equations of motion are solvable with basic differential equations and calculus. A listing of the solutions to both of these first two cases is shown in the summary at the end of this paper. However, for speeds faster than this, the equations of motion have never been solved in closed form. In other words, for the cases of subsonic projectiles where the drag is proportional to the velocity squared and the faster cases involving transonic and supersonic flight, the equations have never been solved.

According to numerous internet sources, the subsonic problem (the case of the velocity squared) was first posed by Isaac Newton some 350 years ago. This problem has become known as "Newton's Riddle" and has stumped mathematicians to the present day (Ref 2). Scientists and mathematicians haveused approximate, numerical methods to solve these equations. While users in the field such as artillery officers and dive bomber pilots have had to rely on thick books of tabulated data to solve their ballistic computations. (Ref 4)

However, certain inroads can be made toward a closed form solution and are most instructive to those interested in ballistics and dynamics. That is the subject of the remainder of this writing and perhaps canbe of some use in the eventual solution of this problem.

Those readers already familiar with the equations of motion may skip the next section. Others may wantto follow through these derivations.

Part 1 Developing The Equations of Motion Those readers already acquainted with the basic ballistic equations of motion may skip this section and proceed to part 2.

For the subsonic case, the atmospheric resistance (drag) is proportional to the square of the speed. Variations in the acceleration of gravity and density of the atmosphere with altitude will be ignored as well as three dimensional effects such as crosswind, coriolis effect, etc. This is a two dimensional problem only. The only forces involved will be the weight of the projectile (w) and the atmospheric drag where the constant of proportionality is k. Therefore, referring to the force diagram, Figure1, and summing the forces in the vertical direction gives.

Figure 1

goes to zero, and the weight becomes equal to the drag. Replacing these values into the previous equation gives.

Then w .

This is the terminal velocity.

Substituting this back gives the vertical equation of motion. (1.)

Similarly for the horizontal direction, the drag is the only force having a horizontal component.

Substituting for the terminal velocity gives the horizontal equation of motion.

(2.)

Then using the same procedure tangent to the flight path gives.

(3.)

This is the acceleration along the flight path.

Then summing the forces perpendicular to the flight path for the centripetal acceleration yields.

(4.)

Then lastly for the conservation of energy, the total mechanical energy at launch equals the mechanical energy at any time afterwards plus the energy lost to drag up to that point in time.

The force F is the drag and is always tangent to the flight path.

Therefore, where s is distance traveled along the ballistic arc.

Therefore, the conservation of energy law says.

or

(5.)

These five equations of motion are restated below and can be described as a set of coupled, second order, non linear, transcendental, ordinary differential equations which in the terminology of mathematicians constitutes an initial value problem. (Ref 3). These equations are well known and can

be found in textbooks dealing with dynamics (Ref 5). These are the ones that must be solved in order to solve a general, subsonic ballistics problem involving drag.

(1.)

(2.)

(3.)

(4.)

(5.)

Part 2 Integration With Respect To Theta

There are several variables in this problem but there is only one independent variable. That is time. Theothers are parameters and are dependent upon the launch scenario (i.e., initial conditions) and time. Since there is only one independent variable, ordinary differential equations will be used.As previously stated, these equations have not been integrated with respect to time but some of the equations can be solved by integrating with respect to theta, the flight path angle above the horizon.

The first four of these equations can all be solved for -g Since all four are equal to -g dt they are all four equal to each other.

(1.) (2.) (3.) (4.)

Putting equations 2 and 4 together gives:

Rearranging

and dividing both sides by cosine squared gives.

Since this last equation can now be integrated.

The solution is:

Canceling and rearranging gives Equation 6, a puzzling and very interesting equation. Upon inspection one would want to call this a "new conservation equation". However, it is not entirely new. It was discovered in another manner and in another form by Shouryya Ray and previously by a few others (Ref 7). However, it does appear to be some type of conservation equation. But what is being conserved? This unitless quantity is not recognizable as anything being taught in physics classes.

(6.)

This "conservation equation" is nonetheless valid for all ballistics in which the drag is proportional to thevelocity squared. A similar equation is shown in the summary at the end of this article for the low speeddrag case.

Being just as valid as the conservation of mass, momentum, angular momentum, and energy, equation 6 also deserves a name. It was decided to give it a temporary name until someone determines what type ofquantity it really is. Since the launch velocity and angle of launch uniquely determine the shape of the ballistic arch, it will be called the "conservation of arch" equation for want of a better name.

The entire right side of this equation is a constant which will be called "A" for arch. It is a small and unitless number for relatively flat, low and fast ballistic trajectories such as rifle fire and becomes much larger for high and slow trajectories such as mortar fire. See Figure 2. Therefore, "arch" appears to be a fairly good name for it since it is a measure of the steepness and amount of arch in the trajectory.

The log produced above, has a special name in mathematics. It is the inverse Gudermannian function (Ref 8). This function relates trigonometric and hyperbolic functions and can take on many forms. They usually involve a composite function with an inverse trig function operating on a hyperbolic function or vice-versa. Consequently, the equations that use it can take on many forms. The symbols gd and gd are commonly used for the Gudermannian and inverse Gudermannian functions. Therefore, the next two equations are equivalent forms of the preceding equation 6.

(Ref 7)

Gudermannian functions occur many times in this reading and the reader should become familiar with them. They may be the key to solving this problem. Since the following integral is long and tedious, it was decided to call it "Is3" for the "Integral of the Secant Cubed". Likewise, the logarithms in the same equation will be called "Is1" because together they constitute the "Integral of the Secant". It is noteworthy that the Is3 integral is a reduction formula integral. This will be disgussed again later.

Substituting Is3 back into equation 6 gives it in a much less complicated form.

(6.1)

This equation can now be solved for , , and v as functions of theta since and

Therefore, , , and v now become defined functions of theta.

But since it is usually more desirable to have only v and theta in the new conservation equation, Equation 6.1 can be rewritten as equation 6.2, since .

(6.2)

Figure 2

The preceeding figure shows a series of level curves of constant arch. This amounts to equation 6.2, the conservation of arch equation, being plotted for several different launch velocities and elevation angles with a terminal velocity of 1000 feet per second. Contrary to high school physics which ignores the effects of drag, it is worth noting that the minimum speed in each trajectory does not occur at the maximum altitude (when passing through level flight). Because of the drag, the velocity will continue todecrease after passing through level flight until the forward component of the weight is equal to the drag (i.e. . Consequently, the line curving up and to the left will be called (for now) the minimum speed line. The minimum point of all ballistic trajectories plotted in this manner must be located on the minimum speed line. It is also noteworthy that these trajectories are not parabolas, nor are they symmetric about the zero climb angle line, and not even the minimum speed line even though they might appear to be so. There is no symmetry. Additionally, it is worth noting that when plotted in this manner all possible ballistic trajectories with a given terminal velocity and launched at or below this velocity will appear on this graph with no two trajectories ever intersecting.

With

and integrating this gives.

Where s is the integral of velocity with time or the arc distance traveled. Continuing, equation 7 is produced which reveals to be a decaying exponential of arc distance traveled.

(7.)

Setting this equal to the equation above for , gives:

The result is compared to the no drag case.

(8.) (no drag)

This brings up an interesting point. Is3 (integral of the secant cubed) appeared above when equation 2 was integrated with equation 4 but yet it reappeared for the case with no drag. This is a result of integration with respect to theta and is discussed below.

The square root above in equation 8 will appear many times in the remainder of this writing. So for the sake of brevity an abbreviation has been assigned to it. Therefore, by letting

the three velocities, in addition to s, can be expressed in terms of L and the

terminal velocity.

It is possible to continue in this manner computing parameters by integrating with respect to theta. Therefore, it will be necessary to use equation 4 to switch the integration variable to theta.

. (4.)

Some time saving relations can now be created

(regardless of drag)

Substituting the value for vx computed above, the relation is made specific to the second power drag case.

(drag =

At this point the appearance of Is3, the integral of the secant cubed, can be explained. In the equation above, it should be noticed that time integrals of various powers of velocity will produce integrals of even higher powers of the secant. Therefore, integrals of the secant cubed are to be expected when n is one regardless of the drag.

It is also noteworthy that integrals of powers of the secant are solved by means of reduction formulas. This means that solutions of these integrals contain integrals of the same function to a lesser power. This will also hold true for the associated time integrals of velocity even though these integrals cannot asof yet be integrated. Therefore, time integrals of velocity raised to a power should also be expected to contain integrals of velocity to a lesser power.

Now for computing the energy lost due to drag from the specific energy equation, n is equal to 3.

(5.1)

It is unlikely that this integral can be evaluated except by numerical means. However, doing so gives excellent results and will allow the vertical distance y to be computed from equation 5.

Time can also be computed using the same relation with v to the zero power. The result is compared with time for the no drag case.

(drag = (no drag)

Finishing with the computation for x gives.

x

Interestingly enough one can see that and

Even though it was necessary to leave x, y, t, and Eslost as unevaluated integrals there are advantages to having done this. Since these integrals are now functions of the integrating variable only, the potential exists for them to one day be evaluated. Also, using numerical techniques to evaluate these integrals will produce a more accurate solution than an iteration scheme to integrate the differential equations. .

It would have been preferable to have all these parameters computed as functions of time but the preceding equations are all solved as functions of theta. However, since time has also been computed as a function of theta, all of the parameters can be plotted versus time.

Part 3 Special Case Scenarios

As previously stated, the general case differential equations have not been solved with respect to time. However, solutions do exist for some special case example problems and are most instructive. They may possibly provide clues to solving the general case equations. These cases are for launches up and down on a frictionless, inclined plane where atmospheric drag is the only friction.

Since theta now becomes a constant, the equations are solvable with ordinary calculus. The turn rate becomes zero and the horizontal and vertical velocities are easily obtainable with the appropriate trig function once the velocity is computed. Therefore, equation 3 now becomes the most important equation to solve.

For launches on a fixed angle, inclined plane equation 3 is solved directly using the separation of variables (Ref 6, p 330&350).

(3.)

(9.)

(10.)

It is instructive to study equation 10 for a moment. The maximum value in the range of the hyperbolic tangent is one. Then equation 10 is saying that the maximum velocity to be seen in this inclined plane example is In other words, the terminal velocity has been reduced by the square root of the sine of the flight path angle (this is negative for the descending case). This makes very good sense since the component force of gravity propelling the projectile forward is also diminished by the same amount. This is also true for the general case ballistic trajectory. This was seen earlier in section 2, figure 2, in which the minimum speed line was computed for the equilibrium condition where the forward component of the weight equaled the drag (i.e. ). Solving for the velocity here will

produce the same term .

This presents the need perhaps for a more in depth understanding of the term "terminal velocity". The terminal velocity is not a barrier that cannot be exceeded. In fact, most projectiles are launched at speeds far in excess of their terminal velocities. In this situation, regardless of the climb or descent angle, the projectile will always decelerate and finally reach its terminal velocity (i.e. equilibrium speed)when going straight down. Actually there is an equilibrium speed of zero acceleration that will exist for each and every descent angle. For any given descent angle, projectiles traveling faster than this speed will be in the process of decelerating toward this speed and slower projectiles will be accelerating toward this speed. So this equilibrium speed (previously called the minimum speed) has been given the symbol for "equilibrium velocity" and can be thought of as a "local" terminal velocity.

It is also instructive to consider equations 9 and 10 for the purely vertical cases of launches straight up

and the inclined plane example and the ballistic trajectory will become identical. Also, since the verticalvelocity is the only velocity, v and vy become equal in magnitude. The equations simplify to the following:

(9.1)

(10.1)

The general case solutions (if and when they are ever solved) must reduce to the above equations for the special case scenarios described above. For example, the limits of the general case v and when goes

to and - must reduce to the special case of 9.1 and 10.1 above.

(9.2)

(10.2)

Therefore, it is reasonable to assume that the general case solution for velocity will be structured similarly to equations 9 and 10 above which are more general than 9.1 and 10.1. This is discussed further in the next section.

The following graph, Figure 3, shows a comparison in vertical velocities between the numerical solution of a horizontal, two dimensional ballistic launch and a launch dropped from rest using equation 10.1. The two comparisons are nearly equal. This tends to indicate that the special case equation for a launch vertically down requires only a small correction to hold true for the two dimensional, general case. Also, this correction must disappear when the limit is taken. This graph and the one following also beliethe teachings of high school physics claiming that a projectile dropped vertically from rest will hit the ground at the same time as one fired horizontally. Figure 4 shows a series of launches at different horizontal velocities. When drag is considered, a projectile with any horizontal velocity will take longer to hit the ground than one dropped from rest.

Figure 3

Figure 4

Part 4 Clues and FindingsWhile being extremely complex and so far unsolvable, the mathematics of Newton's Riddle does reveal some clues that should be noticed and hopefully be expanded upon to eventually provide a solution. Trig functions, hyperbolics, exponentials, reduction formulas and Gudermannian functions all continuously reappear and will probably appear in the final answer. If an answer cannot be computed in closed form from conventional mathematical procedures perhaps a solution can be pieced together from knowledge of the problem and a series of clues the mathematics has provided.

Clue number 1.

Using equation 4 one can see that It is noteworthy that drag (i.e. vt) is not

mentioned in this equation.

Is1 =

a nice decaying exponential.

(4.2)

Alternately, it could be solved in terms of Gudermannians.

(4.3)

This is not a complete solution but it is enough to show that in all ballistic trajectories theta will be a composite, Gudermannian function of time with Gudermannian as the outer function. This also means that trig functions of theta will be hyperbolic functions of time (Ref 6). These two conclusions will not change whether drag is considered or not.

Clue number 2.

Since another form of the Gudermannian function is , equation 4.3 can be converted to the following:

. (4.4)

and since it can be seen that this looks

suspiciously like equation 10.1.

(10.1)

This also tends to indicate that the general case (closed form) solution for might be very similar to the one shown for (down), and possibly a Gudermannian function. However, it must be remembered that the other special case scenario solved in part 3, gave this equation for (up).

(9.1)

A true solution to this problem must accommodate both the ascending as well as the descending phases of flight. So is there a type of function that can be adaptable for the ascending (tangent) case as well as the descending (hyperbolic tangent) case? Yes, there is. The following identities are among several which relate circular trig functions and hyperbolic functions.

(Ref 6, p 256)

These functions indicate that when the tangent argument becomes imaginary the function becomes hyperbolic. Equations 9 and 10 already fit these identities for ascending as well as descending flight. Referring to equation 9, for descending flight, and the argument inside the tangent brackets both become imaginary and consequently become equivalent to equation 10. This tends to indicate that equations 9 and 10 may be very close to a solution for the general case problem, equation 3, keeping in mind that they were derived considering theta to be a constant.

If theta is considered to be a variable in equation 3, as it most definitely is in reality, a number of difficulties arise.

(3.)

In order for this equation to be solved, v and theta must be either known functions of time which is precisely what is being sought or theta must be a known function of v. In section 2 of this paper, v was successfully solved as a function of theta but that equation cannot be inverted and solved for theta as a function of v.

Assuming, for the moment, that the sine of theta is some known function of v , it is instructive to consultintegral tables searching for integrals of the following form in hopes they might tend to indicate some general characteristics expected in an actual solution.

Most of the integrals found, which were few in number, involved multiple occurrences of v in transcendental equations that could not be solved for v. That possibility should be considered as it

would frustrate attempts to take a closed form solution any further.

Clue number 3.

Using equation 4.4equation 3 is produced.

(3.1)

One might tend to think that this substitution makes the equation even more complicated and unsolvable but it does reduce the problem down to one dependent variable (v) and one independent variable (t) which is a valuable and necessary step forward. In this form it might possibly be solved. However, it is still a nonlinear, transcendental, integral/ differential equation. So textbooks provide little guidance into solving equations as complicated as this.

Part 5 Recommendations for Further Work

It is felt that conventional mathematical procedures will not be fruitful in solving this problem especiallythose manual in nature. The calculations are long and tedious and are consequently prone to clueless dead ends and errors. An automated method of using the power of the computer with symbolic math to gradually piece together a solution from lessons learned has certain promise. Working a series of increasingly complex example problems was shown to be a valuable learning aid and this concept shouldbe continued.

Equations 9 and 10 show a lot of promise toward the solution of the velocity equation. Using these two equations as a basis they should be expanded upon. The accelerations produced by differentiating these two equations come close to solving the velocity equation and tend to confirm that these efforts are proceeding in the right direction.

Gudermannian functions continuously appear and reappear in the calculations. It was shown earlier in part 4 that theta will be a Gudermannian function of time with suspicions that v and might be the same. Since these functions are so adept at providing the links between circular and hyperbolic functions, they should be studied more in depth.

Finally, the conservation of arch needs to be studied in greater detail for it appears to be a new quantity of interest in physics.

Part 6 Conclusion

This problem is far from solved. However, it is felt that some new and significant advancements have been achieved here.

This paper has:

1. Produced an equivalent equation to the one produced by Shouryya Ray, but by a different method.

2. Identified this equation as a conservation equation.

3. Used this equation to compute v, , , and s as defined functions of theta.

4. Named and graphed level curves of the "Arch Equation".

5. Computed the remainder of the ballistic parameters as unevaluated integrals of theta.

6. Identified Is3 as a quantity of interest in ballistic physics.

7. Studied and solved special case scenarios.

8. Defined and studied the equilibrium velocity and showed its relationship to the terminal velocity.

8. Identified and used Gudermannian functions to aid in possible solutions.

9. Reduced the velocity equation to one dependent and one independent variable.

10. Identified the basic forms of expected solutions.

11. Recommended areas for future study and work.

Part 7 A Summary of Solvable Ballistic Equations

Drag = 0 Drag = k x v

= A

Part 8 Acknowledgements

The author would like to thank Dr. Charles K. Cook, retired professor of mathematics at the University of South Carolina (Sumter Campus) for his suggestions to help make this paper more presentable to the general reader.

Part 9 References (In order of appearance)

1. Thomas, Calculus and Analytic Geometry, Addison - Wesley Publications, 1962, pp. 561-565.

2. Newton's Riddle, http;//www.ottawacitizen.com/technology, May 28,2012

3. Shampine, Solving ODE's With Matlab, Cambridge Univ. Press, pp 12-13.

4. USAF Weapons Delivery Computations, Tech. Order 1F-4C-34-1-2

5. Langhaar & Boresi, Engineering Mechanics - Dynamics , McGraw Hill, 1959, pp. 433-441.

6. CRC Standard Math Tables, 25th edition, pp 262-264.

7. Dr. Ralph Chill, Shouryya Ray internet article, Technical Univ. of Dresden, June 4, 2012, http://tu-dresden.de/

8. Wolfram - Gudermannian function, http://mathworld.wolfram.com/gudermannian.html.