A Method for Seeking Range of Missing Plane Based on Law of Energy Conservation

A method for seeking range of a missing plane based on Law of energy conservation is suggested. The method starts with the final information of the location(s) of the plane A(x,y,z) provided by satellites. A flying path with most far from an origin in same time interval [0,T] is proved. Suppose that the flight flies in non-powered state since A(x,y,z). By the Law of energy conservation, the problems corresponding to two ranges of velocity are reduced to Riccati equation and Volterra equation respectively. The exact solutions of these equations are found, and the seeking range is calculated. Finally, the relative motion in longitude direction due to self-rotation of earth is added to the calculated range. A simple calculation is given for reference.


INTRODUCTION
Missing flights e.g., MH370 (2014/03/09 1:07am), QZ8501 (2014/12/28 7:24 am) are widely concerned. Seeking missing flight needs cooperation with international community in three steps: 1, tracking flight with satellites and radar stations, 2, determination the range of seeking. 3, pickup parts of the plane. As an example of application of [1], this short note tries to determine a maximum range of the plane possible dropping based on the law of conservation energy.
The law of conservation energy is one of the greatest discovers of 20's century in the world. This law has been widely used in many fields and widely accepted as a criterion for scientific judge on true or false. For example, the law announces that none of numerous designs of perpetual motion machine in centuries becomes true.
Energy method based on principle of work and energy, conservation law of energy, etc., has been widely used in mechanics. Comparing with the differential equation method, energy method is more welcome by users of practical application. They do not care the description of the whole process, i.e., the solution of the differential equation, while they are interested in some special points, e.g., the turning points, the equilibrium points, the maximum, minimum points, etc. Energy method can achieve to these goals easily. Obviously, to seek a missing plane, we are interested in the largest range of the plane possible dropping, not the details of the actual path, so that the method of energy is suited for our goal. At first, according to [2][3][4], the plane probably exhausted its fuel and dropped in Indian Ocean. We make an assumption that the missing flight flied in non-powered state since A(x,y,z). Then, we prove an assertion that amount numerous paths, the path flying most far from an origin in the same time interval [0,T] must be flying along a straight line, in section 2.
In section 3, two integral equations involve unknown velocity function, i.e., the Riccati equation and Volterra equation, are set up based on the law of conservation energy corresponding to two ranges of velocity. The exact solutions of these equations are found, and the seeking ranges are calculated.
In section 4, a relative motion in longitude direction in time interval [0,T] due to self-rotation of earth is added to the maximum of the two calculated ranges to form the final range. In section 5, a simple calculation is given for reference. Finally, a conclusion is made.
Note that more general models of the plane dropping can be formulated in the case of more general flight phenomena. However, in the case considered in this paper, the model based on the law of conservation of energy appeared to be sufficient.

Coordinate System
Let A ( x,y,z) be the location of final information on the plane provided by satellites and radar stations in step 1, where coordinates x, y, z are axes along longitudinal, latitudinal and vertical direction respectively.
Let the calculating coordinates of the plane be A (r, θ , h) in step 2, with origin at A(0,0,0) = A (x,y,z) the location of the plane in air. Where the polar coordinates x = rcos θ, y = rsin θ, and h = -z, the vertical h-axis is in opposite direction with the z-axis.
Suppose that the plane at location A (r,0, h) flying along a s-axis, has an increase ds at time t, the relations between the s-axis and r-axis, the saxis and the h-axis are: The relation between componentsdr, dh and ds is: is the angle between the r-axis and the s-axis. where

Proof of the path of the flight possible dropping with maximum radius, centered at A (0,0,0)
According to the news report from CNN 2014-03-15 that the flight MH370 probably exhausted its fuel dropping in Indian Ocean [2], suppose that the flight flied in a non-powered flying since A(0,0,0) = A(x,y,z).
There are numerous paths of the plane flying. We need not know and we are impossible to know the actual path of the plane did. We are interested in one path that goes most far from A(0,0,0) = A(x,y,z) in the same time interval [0,T].
Let R = R(t) = max∫ v (θ, α)dt be the maximum radius of the plane dropping, centered at A(0,0,0). Where θ = θ(t) is the angle between plan rOh and plan xOh, α = α(t) is the angle between the s-axis and the r-axis in plan rOh. Suppose that the plane flies along the s-axis with velocity v(t) at time t. Where The necessary condition of R to be an extreme v is a function of θ and α, (2-7) give: dθ/dt = 0, and dα/dt = 0. (2)(3)(4)(5)(6)(7)(8) i.e., θ= C 1 . And α = C 2 , C 1 and C 2 are constants. That means the plane flies along the s-axis in plane rOh with constant angle α = C 2 in any time t ∈ [0, T], i.e., to get a maximum distance from A (0,0,0) in the same time interval [0,T], the flight must fly along a straight line, say a s-axis.
In the following, the plane flying along the s-axis, i.e., α = C 2 (constant), is discussed.

SET UP VELOCITY EQUATION BY THE LAW OF CONSERVATION ENERGY
For the case of non-powered flying, there are three forces applied to the plane, i.e., the drag F , the lift F and the gravity force w.
The formulas of drag F and lift F are varied with velocity v(t) [5] (notice the difference between the following (3-1),(3-2) and (3-18), (3-19)). In the following, since the super-sound speed and sound speed are impossible for a non-powered flying, so that two possible ranges of velocity, i.e., the sub-sonic speed and low speed are discussed.

The Velocity of the Plane Under Subsonic Speed
In which the formulas of drag F and F are usually expressed [5] by (3-1) and (3-2) Where v = v(t) denotes velocity along the s-axis at time t; k and k are constants.
These simple formulas are suited for average case with enough accuracy. If one needs complex formulas with more accuracy on drag and lift, one can refer to [6].
where m is the mass of the plane; g = 9.81 (m/sec 2 ) is the gravity acceleration. Drag F applies along the s-axis with opposite direction of ds; F applies perpendicular to the saxis. The w applies along the h-axis (z-axis). During the process of the plane flight along the saxis, its displacement from 0 to s, no work has been done by force F , so that the mechanical energy E(t) is: Where the first, second and last terms of the right-hand side denote the work done by w, by
The solution of this special kind of Riccati function is (formula 1.23 of [3]): , for AB > 0, Substituting (3-1), (3-2) into (3-10) and (3-11), we have Where the coefficients k and k can be found by flight test, by wind tunnel test, for a wished speed. A higher lift-to-drag ratio (L/D ratio) is typically one of the major goals in aircraft design. It leads directly to better fuel economy, climb performance, and glide ratio. According to data (3-17) shows the largest radius R of the plane possible dropping with centered at A(0,0,0)=A(x,y,z).

The Plane Flight with Low Velocity
In the case of a low flight velocity, the drag F and lift F are expressed by (3-18) and (3-19).
Where k , k are constants.
Similarly to section 3.1 above, By the law of conservation energy, E(0) = E(t), we have: (3-21) is a non-linear integral equation with unknown functions h, v, v. Using  and v(t) = ds/dt , (3-21) becomes a non-linear Volterra integral equation of the second kind [8] and can be easily solved.
Differentiating both sides of (3-21) respect to t, we have: (3-33) shows the largest radius R of the plane possible dropping with centered at A (0,0,0) = A (x,y,z).

ADDITIONAL RELATIVE MOTION DUE TO EARTH SELF-ROTATION
In the following, the earth is considered as a sphere with a radius R ⊕ = 6371 km, self-rotating around north-south polar axis with angular velocity ∆∅ = 360 o /24h = π/12 (radian/hour) = π/720 (radian/min).
The displacement Δx of point A(x,y,0) in time [0,T] due to self-rotation of earth is: Where T represents the time from A(x,y,z) to dropping into ocean. R ⊕ cosy represents the radius of the latitude circle at y.
Since the plane is flying in air, not affiliated with the earth, so that the self-rotation of earth does not influence on the plane. However, when the plane dropping into the sea, the calculated dropping point on earth has moved a distance Δx, so that Δx should be added into each boundary point of the calculated range. For simple, Δx is added to the center A (x,y,0), i.e., A (x+Δx,y,0). The final calculated range is a circle, centered at A (x+Δx,y,0) with radius R ((3-17) or (3-33)), depends on the initial velocity v 0 belonging to under sub-sound speed or low speed.
There is expert evaluate that the influence on the final result due to earth self-rotation is about 6%.
If several initial points are provided by satellites, then, repeated the above calculation for each initial points.

A SIMPLE CALCULATION OF THE RANGE OF POSSIBLE DROPPING
The above calculation would be complex. One needs a simple estimation for reference.

CONCLUSION
Suppose that the flight flied in a non-powered flying since A(0,0,0) = A(x,y,z).
There are numerous paths of the plane flying. The path which goes most far from A(0,0,0) must be flying along a straight line in the same time interval [0,T].
Two integral equations involve unknown velocity function, i.e., the Riccati equation and Volterra equation, are set up based on the law of conservation energy corresponding to two possible ranges of velocity. The exact solutions of these equations are found, and the seeking ranges are calculated. In which the angles of depression tan α = F s / F L (3-12) or (3-27), depending on the ratio of drag to lift corresponding to two possible ranges of velocity respectively.
Finally, The final calculated range is a circle, centered at A(x+Δx,y,0) with radius R ((3-17) or (3-33)), depends on the initial velocity v 0 belonging to under sub-sound speed or low speed. We can choose the maximum one of R for seeking.