Method for calculating the trajectory of an object dropped into water from the air

Abstract

This invention discloses a method for calculating the trajectory of an object dropped from the air into water, and for calculating the drag force F in the direction of water flow y. D The upward force f in the direction of water depth z D This invention establishes the state equations for a bomb falling through the air until it just touches the water surface, calculates the horizontal displacement of the bomb in the air, and sums the horizontal displacements in the air and sea to obtain the relative horizontal displacements between the aircraft and the submarine. Using a dynamic approach and considering the influence of various objective factors, this invention derives the trajectory equations of an object entering the water through dynamic differential equations and boundary conditions. These trajectory equations can be generalized.

Description

Technical Field

[0001] This invention belongs to the field of military training technology, and in particular relates to a method for calculating the trajectory of an object dropped from the air into water. Background Technology

[0002] In modern warfare, militaries worldwide often employ joint operations to conduct three-dimensional warfare. In modern joint operations, aircraft bombing is frequently used to attack moving targets on the surface and underwater. Actual training aims to more effectively destroy targets by analyzing factors such as aircraft speed, altitude, and direction; target position, size, speed, and direction of movement; and bomb size, weight, wind speed and direction at sea, and sea density and depth to determine and calculate the accuracy of bombing. The problem states that the bomb is approximately spherical, has no propulsion, and explodes upon contact with the target. The closer the point of impact is to the center of the target, the greater the damage. Summary of the Invention

[0003] In view of this, the present invention proposes a method for calculating the trajectory of an object dropped from the air into the water. The submarine is regarded as a cylinder. When the bomber drops bombs from high altitude, there will inevitably be a range that cannot be reached. The method analyzes the process from when the bomb just touches the sea surface to when it sinks into the sea, performs force analysis, calculates the time of the bomb's movement in the sea, and thus obtains the trajectory in three directions. After synthesis, the optimal timing for bombing is obtained when the distance between the aircraft and the submarine is multiplied.

[0004] To achieve the above-mentioned objectives, the first aspect of this invention discloses a method for calculating the trajectory of an object launched from the air into water, comprising the following steps:

[0005] Calculate the drag force F in the direction y of the water flow. D The upward force f in the direction of water depth z D :

[0006]

[0007]

[0008] Calculate the effective gravity P:

[0009] P = mg - ρ²gV

[0010] In the formula, C D C represents the drag coefficient in the direction of seawater flow, i.e., the drag force coefficient. Lρ is the drag coefficient in the downward direction, i.e., the lifting force coefficient; m is the mass of the bomb, m = ρ0V; g is the acceleration due to gravity; ρ2 is the density of water; ρ0 is the density of the bomb; u0 is the velocity of the water flow around the bomb; u is the velocity of the object in the direction of the water flow; ω is the downward velocity of the object in the direction of water depth; A h A is the equivalent area of ​​the horizontal cross section; P d is the equivalent area of ​​the cross section in the numerical direction; V is the volume of the object, and d is the diameter of the bomb.

[0011] Establish the equation of state for the bomb falling through the air until it just touches the water surface:

[0012]

[0013] in have Integrating over time t, we have the equation:

[0014]

[0015] The state of the bomb when it enters the water:

[0016] Considering the initial velocity in the z-direction when entering the water, we get:

[0017]

[0018]

[0019] Substituting β1 and β2 into the above equation, we have β1 = 1 / u0, β2 = 0. Substituting these values ​​back into the equation, we get:

[0020]

[0021] Perform an equation transformation, let

[0022] Integrating over time t and rearranging, we obtain the system of equations:

[0023]

[0024] Substitute u0 and A h We obtain the time t and the horizontal displacement y of the bomb in the seawater;

[0025] The horizontal displacement of the bomb in the air can be calculated using the following formula:

[0026]

[0027] Among them, v 0y Let be the initial velocity of the bomb in the y-direction in the air. Summing the horizontal displacements in the air and sea, we obtain the relative horizontal displacements between the aircraft and the submarine.

[0028] The second aspect of this invention discloses a method for calculating the trajectory of an object launched from the air into water, comprising the following steps:

[0029] According to Newton's second law, the bomb satisfies the following differential equation:

[0030]

[0031] v y Let u0 be the absolute velocity in the vertical direction in the water, and c be the velocity of the water flow around the bomb. D ρ is the drag coefficient in the direction of seawater flow, i.e., the drag force coefficient; m is the mass of the bomb; ρ2 is the density of water; u0 is the velocity of the water flow around the bomb;

[0032] The corresponding initial conditions are: when t = 0, y = 0, and when dy / dt = vy = 0, the equation is:

[0033]

[0034] Integrating both sides yields

[0035]

[0036] in A is the integration constant. Substituting the initial condition at t=0, dy / dt=vy=0, we get...

[0037]

[0038]

[0039]

[0040] Finally, integrating both sides with respect to t yields the result.

[0041]

[0042] Given the initial condition t = 0, y = 0, we can solve for B = 0. Therefore:

[0043]

[0044] From the moment the bomb touches the water surface until it is just fully submerged, the bomb experiences buoyancy, water resistance, and gravity along the z-axis. The buoyancy is:

[0045]

[0046] c is the drag coefficient;

[0047] The effect of gravity on the speed of a bomb is as follows:

[0048] v z重=v0+gt

[0049] Where g is the acceleration due to gravity;

[0050] The effect of buoyancy on bomb speed:

[0051]

[0052] In the formula S z It represents the bomb's area along the Z-axis.

[0053] Taking into account the effects of gravity, buoyancy, and water resistance, the following equation is obtained through velocity composition:

[0054]

[0055]

[0056]

[0057] During the stage from when the bomb is just fully submerged in the water to when it sinks to the bottom,

[0058]

[0059] Substituting the initial condition at t=0, z=-c / 2, we get

[0060] In summary, the parametric equations for the bomb's motion throughout the entire descent process are as follows:

[0061]

[0062] Where d is the diameter of the submarine.

[0063]

[0064] Substituting the corresponding parameter values ​​into the above model, the horizontal displacement of the bomb in the seawater is obtained. The horizontal velocity and displacement of the bomb in the air are calculated. The horizontal displacements in the air and in the sea are summed to obtain the horizontal relative displacement between the aircraft and the submarine and the vertical displacement between the aircraft and the submarine.

[0065] Furthermore, when the target submarine's possible diving depth is within a certain range, the following model is established:

[0066]

[0067] The beneficial effects of this invention are as follows:

[0068] This invention employs a dynamic approach, taking into account the influence of various objective conditions. Through dynamic differential equations and boundary conditions, it obtains the trajectory equation of an object entering water, and this trajectory equation can be generalized. Attached Figure Description

[0069] Figure 1 Force diagram at the center of the bomb of this invention;

[0070] Figure 2 The relationship between displacement in the y-direction and time;

[0071] Figure 3 Relationship between displacement in the z-direction and time. Detailed Implementation

[0072] The present invention will be further described below with reference to the accompanying drawings, but this is not intended to limit the present invention in any way. Any modifications or substitutions made based on the teachings of the present invention shall fall within the protection scope of the present invention.

[0073] To simplify the problem, this invention assumes a bomber with a flight speed of 200 m / s and a flight altitude of 1000 m; a bomb with a radius of 0.25 m and a density of 0.8 × 10³ kg / m³; and a wind speed at sea of ​​10 m / s. The target is an approximately cylindrical submarine with a length of 100 m, a maximum width of 9 m, a diving depth of 30 m, and a speed of 15 knots.

[0074] This invention models and analyzes the relationship between the accuracy of aircraft bombing and various related factors. It determines the optimal timing for aircraft bombing for three targets under different conditions: the direction of the aircraft's movement is the same as, opposite to, perpendicular to, and at a 45° angle to the target. The model's results are then simulated and verified.

[0075] In this invention, the submarine is cylindrical with a certain thickness. Therefore, the bomb can hit not only the front of the submarine but also part of its sides. This increases the effective bombing area for the aircraft in the air, and the effective vertical travel of the bomb in the water will increase by the diameter of the cylinder. This must be considered during the model building process.

[0076] Example 1

[0077] This embodiment needs to consider the submarine's depth in the sea; the problem treats the submarine as a cylinder, and when the bomber drops bombs from high altitude, there will inevitably be a range that it cannot reach. Based on the bomb being dropped to the sea level, the process from the moment the bomb touches the sea surface to when it sinks into the sea is analyzed, force analysis is performed, and the time of the bomb's movement in the sea is calculated, thereby obtaining the trajectory of the bomb in three directions. After synthesis, the optimal timing for bombing is determined when the distance between the aircraft and the submarine is multiplied.

[0078] When a bomb enters water from the air, it will be affected by seawater resistance and impact during its descent, causing it to drift. Assuming the underwater submarine is stable and the bomb is a point mass, we only consider buoyancy, gravity, and the drag force F from the current. D Dragging force is the pulling force that slows down a bomb in the horizontal direction, while the lifting force f in the direction of water depth is the force that increases the speed of the bomb. D The influence of water on the forces acting on an object. Figure 1 As shown.

[0079] Combining gravity and buoyancy into an effective gravity P, calculate the drag force in the y-direction of the water flow according to fluid mechanics:

[0080]

[0081] Uplift force in the z-direction of water depth:

[0082]

[0083] The contact area between the bomb and the water is πr. 2 Substitute the area into the above formula to calculate the drag force and lifting force:

[0084]

[0085]

[0086] Effective gravity

[0087] P = mg - ρ²gV

[0088] The mass of the bomb is: m = ρ0V

[0089] In the above formula, C D C represents the drag coefficient in the direction of seawater flow, i.e., the drag force coefficient. L ρ is the drag coefficient in the downward direction, i.e., the lifting force coefficient; m is the mass of the bomb; g is the acceleration due to gravity; ρ2 is the density of water; ρ0 is the density of the bomb; u0 is the velocity of the water flow around the bomb; u is the velocity of the object in the direction of the water flow; ω is the downward velocity of the object in the direction of water depth; A h A is the equivalent area of ​​the horizontal cross section; P V is the equivalent area of ​​the cross section in the numerical direction; V is the volume of the object.

[0090] Swaying force in the y direction:

[0091]

[0092] Resultant force in the z-direction:

[0093]

[0094] F Df D Substituting the expressions for P and m, we have

[0095]

[0096] make The unit is 1 / m. The unit is m / s. The unit is 1 / m, which obviously means that Integrating over time t, we have the equation:

[0097]

[0098] The equation solves for the state of the bomb as it falls through the air until it just touches the water surface, which is the critical state. The following discussion addresses the state when the bomb enters the water:

[0099] Considering the initial velocity in the z-direction when entering the water, we can obtain:

[0100]

[0101]

[0102] Substituting β1 and β2 into the time equation above, we get β1 = 1 / u0 and β2 = 0. Substituting these back into the equation, we get:

[0103]

[0104] Perform an equation transformation, let

[0105] Integrating over time t and rearranging, we obtain the system of equations:

[0106]

[0107] Based on the above model, substitute the corresponding parameter values:

[0108] u0 = 0.34 m / s

[0109]

[0110] Substituting the parameters into the equation yields the following solution:

[0111] Since the initial velocity in the x-direction is small, it experiences almost no deflection after entering the water due to resistance, so this can be ignored. t = 2.8s, solving for t gives:

[0112] The horizontal displacement of the bomb in the seawater: y = 0.512m.

[0113] The horizontal velocity and displacement of the bomb in the air can be calculated using the following formula:

[0114]

[0115]

[0116] Among them, v 0y Let m be the initial velocity of the bomb in the y-direction, m be the mass of the bomb, and k be the air resistance coefficient.

[0117] v y =0.6045m / s, displacement y =63.2495m.

[0118] Summing the horizontal displacements in the air and sea, we obtain the relative horizontal displacement between the aircraft and the submarine, y = 63.678 m.

[0119] Vertical displacement of the aircraft and submarine: z = -1050m.

[0120] Example 2

[0121] Establishment of the differential equation of bomb motion

[0122] We take the x-axis as the positive direction of the water flow and the y-axis as the positive direction of the vertical direction. The origin of the coordinate system is the point where the bomb enters the water. The movement of the bomb in the water is divided into two phases: from the bomb's initial contact with the water surface to its complete immersion and from its complete immersion to its sinking. Therefore, when building the model, the x-direction can be approximated as being exactly the same in both phases and does not need to be considered separately. However, the y-direction movement differs in the two phases, so it needs to be discussed separately.

[0123] 1) The distance the bomb travels along the y-axis

[0124] The bomb is only subjected to the thrust of water in the x-axis direction, so the thrust magnitude is constant in the x-axis direction regardless of stage.

[0125]

[0126] Where μ0 is the velocity of the water flow, v x It is the absolute velocity of the bomb in the horizontal direction in the water. In this invention, at the moment the bomb just enters the water, its initial velocity in the horizontal direction is considered to be equal to v0, while the horizontal water flow velocity is regarded as a constant value set as u0 (0.34m / s). The bomb accelerates under the action of thrust until its velocity is equal to the water velocity, at which point the bomb is considered to start moving at a constant speed.

[0127] According to Newton's second law, the bomb satisfies the following differential equation:

[0128]

[0129] The corresponding initial conditions are: when t = 0, y = 0, and when dy / dt = vy = 0, the equation is:

[0130]

[0131] Integrating both sides yields

[0132]

[0133] in A is the integration constant. Substituting the initial condition at t=0, dy / dt=vy=0, we get...

[0134]

[0135]

[0136]

[0137] Finally, integrating both sides with respect to t yields the result.

[0138]

[0139] Given the initial condition t = 0, y = 0, we can solve for B = 0. Therefore:

[0140]

[0141] 2) The distance the bomb travels in the z-axis direction can be considered in two stages because the bomb enters the water in two stages.

[0142] Phase 1: The bomb touches the water surface until it is just fully submerged.

[0143] During this phase, from the moment the bomb touches the water surface until it is just fully submerged, the bomb experiences buoyancy, water resistance, and gravity along the z-axis. Buoyancy...

[0144]

[0145] The corresponding initial conditions are as follows: when t = 0, z = c / 2, dz / dt = v0. Since the effects of gravity, buoyancy and drag on the speed of the bomb are independent of each other, the above equation can be decomposed into three separate forces for solving first, and then synthesized.

[0146] (1) The effect of gravity on the velocity of the bomb

[0147] v z重 =v0+gt

[0148] Where g is the acceleration due to gravity.

[0149] (2) The effect of buoyancy on the speed of the bomb

[0150] According to Newton's second law

[0151]

[0152] Differentiating both sides with respect to time t, we get

[0153]

[0154] This is about v z The second-order differential equation with constant coefficients is solved to obtain

[0155]

[0156] In the formula

[0157] (3) The effect of drag on bomb velocity

[0158] According to Newton's second law, we know that...

[0159]

[0160] Separating variables yields Integrating both sides with respect to time t, we can obtain

[0161] in

[0162] B is the integration constant.

[0163] Taking into account the effects of gravity, buoyancy, and water resistance, the following equation can be obtained through velocity synthesis:

[0164] v z =v z重 -v z浮 -v z阻

[0165] Substituting the results of the above derivation, we have

[0166]

[0167] Based on the initial velocity condition t=0, v z =v0, substituting into the above equation, we get B = -1 / (C+D), integrating both sides of the above equation with respect to t.

[0168]

[0169] Based on the initial position condition t=0, y=c / 2

[0170] achievable

[0171]

[0172] Where B = -1 / (C+D), and C and D are integration constants.

[0173] Phase 2 bombs were completely submerged in the water until they sank to the bottom.

[0174] During this phase, the bomb is subjected to the combined effects of buoyancy, drag, and gravity along the y-axis, with gravity G = mg and buoyancy F... 浮 =ρgV, where V is the volume of the heavy bomb, and V = the water-facing area S × the thickness of the vertical water-facing surface;

[0175] Resistance F 阻 =C D S z ρv 2 =C D S z ρv y 2 ,

[0176] Where v y It is the absolute horizontal velocity of the bomb in the water.

[0177] According to Newton's second law, the bomb satisfies the following differential equation at this stage.

[0178]

[0179] The corresponding initial conditions are: t = 0, z = -c / 2

[0180] Since the function is continuous at the point of segmentation in both the preceding and following stages, at t=0, The buoyancy is equal in magnitude to that at the same location in the previous stage. Compared to the previous stage, the buoyancy is constant in this stage. Therefore:

[0181]

[0182] In the formula but

[0183]

[0184] Substituting the initial condition at t=0, z=-c / 2, we get In summary, the parametric equations for the bomb's motion throughout the entire descent process are as follows:

[0185]

[0186] Where C and D are integration constants, and d is the diameter of the submarine.

[0187]

[0188] 1. Solving the differential equations of bomb motion

[0189] Based on the above model, substitute the corresponding parameter values:

[0190] Given t = 2.3s, u0 = 0.34m / s, c = 0.5m, C = 2.415, and B = 72, we can solve for the horizontal displacement of the bomb in the seawater.

[0191] y = 0.429m

[0192] Find the horizontal velocity v of the bomb in the air. y =0.6045m / s, displacement y =63.2495m.

[0193] Summing the horizontal displacements in the air and sea, we obtain the relative horizontal displacement between the aircraft and the submarine: y = 63.678 m.

[0194] Vertical displacement of the aircraft and submarine: z = -1050m.

[0195] Further analysis of the bomb's descent process

[0196] Horizontal distance change over time

[0197] In the horizontal direction, the bomb has an initial velocity upon entering the water, providing propulsion. Due to water resistance, its velocity gradually decreases until it reaches zero. At t = 8s, the bomb is moving at a constant velocity in the y-direction. In the x-direction, it is only affected by the impact force of the water flow, so the bomb continues to move in the y-direction. This is achieved through fitting data using a MATLAB program, as shown below. Figure 2 As shown.

[0198] 1. The relationship between vertical distance and time

[0199] The bomb is subjected to gravity and buoyancy in the vertical direction. After entering the water, its descent speed gradually slows due to drag, eventually coming to a stop at a depth of 6 meters. Then, because the density of water is greater than that of the bomb, the bomb eventually rises to the surface due to buoyancy. This was achieved through fitting data using a MATLAB program. Figure 3 As shown.

[0200] The aircraft flies in the same direction as the submarine. The best time to drop bombs is when the submarine's coordinates relative to the aircraft are (x1, y1+1.8, -30) to (x1, y1+2.5, -30). The best bombing area for the aircraft is (x1, y1-58.5, -30) to (x1, y1+41.7, -30).

[0201] The aircraft flies in the opposite direction to the submarine. The optimal time to drop bombs is when the submarine's coordinates relative to the aircraft are (x1, y1+1.8, -30) to (x1, y1+2.5, -30). The optimal bombing zone for the aircraft is (x1, y1-38.05, -30) to (x1, y1+62.85, -30).

[0202] The aircraft's flight direction is perpendicular to the ship's direction of travel. The optimal time to drop bombs is when the ship's coordinates relative to the aircraft are (x1, y1+1.8, -30) to (x1, y1+2.5, -30). The optimal bombing zone for the aircraft is (x1, y1-60.8, -30) to (x1, y1+39.65, -30).

[0203] When the aircraft's flight direction is at a 45° angle to the ship's direction of travel, the optimal time to drop bombs is when the ship's coordinates relative to the aircraft are (x1+2.2, y1+1.8, -18) to (x1+2.2, y1+2.5, -18); the optimal bombing zone for the aircraft is (x1-52, y1, -30) to (x1+28, y1, -30).

[0204] Example 3

[0205] In this embodiment, the target submarine may dive to a depth of 20–60 m, and other parameters are the same as in embodiments 1 and 2.

[0206] Model establishment and solution

[0207] Because the density of a bomb is less than that of seawater, if the bomb does not explode, its motion will eventually return to the surface. Therefore, the bomb should sink first, gradually slowing down. Even when its vertical velocity reaches zero, it still experiences an upward buoyant force, causing it to begin returning to the surface. According to the formula...

[0208]

[0209] The trajectory of the bomb in the vertical direction over time was obtained. At the 6th second, its velocity dropped to 0 and it began to rise. Therefore, the bomb is effective when the submarine sinks to a depth between 20m and 50m; when the submarine sinks to a depth greater than 50m, the bomb has no power of its own and is only subject to buoyancy, so it is ineffective against the submarine.

[0210] 1) The aircraft's flight direction is the same as the submarine's direction of travel. The optimal time to drop bombs is when the submarine's coordinates relative to the aircraft are between (x1, y1+0.5, -20) and (x1, y1+5, -50). The optimal bombing zone for the aircraft is (x1, y1+0.5, -20).

[0211] -59.343, -20) to x1, y1 + 42.76, -50).

[0212] 2) The aircraft flies in the opposite direction to the submarine. The best time to drop bombs is when the submarine's coordinates relative to the aircraft are (x1,y1+0.5,-20) to (x1,y1+5,-50). The best bombing area for the aircraft is (x1,y1+40.657,-20) to (x1,y1-22.76,-50).

[0213] 3) The aircraft's flight direction is perpendicular to the ship's direction of travel. When the ship's coordinates relative to the aircraft are (x1+...

[0214] The optimal time to drop bombs is between (x1+2.5,y1,-30) and (x1+2.5,y1,-30); the optimal bombing zone for the aircraft is between (x1-29.65,y1,-30) and (x1+60.8,y1,-30).

[0215] 4) When the aircraft's flight direction is at 45° to the ship's direction of travel, the optimal time to drop bombs is when the ship's coordinates relative to the aircraft are (x1+15,y1+5,-18) to (x1+15,y1+5,-18); the optimal bombing zone for the aircraft is (x1-30,y1,-30) to (x1+18,y1,-30).

[0216] The beneficial effects of this invention are as follows:

[0217] This invention employs a dynamic approach, taking into account the influence of various objective conditions. Through dynamic differential equations and boundary conditions, it obtains the trajectory equation of an object entering water, and this trajectory equation can be generalized.

[0218] As used herein, the term "preferred" is meant as an example, illustration, or illustration. Any aspect or design described herein as "preferred" need not be construed as being more advantageous than other aspects or designs. Rather, the use of the term "preferred" is intended to present the concept in a specific manner. As used in this application, the term "or" is intended to mean an inclusive "or" rather than an exclusionary "or." That is, unless otherwise specified or clear from the context, "X uses A or B" naturally includes either of the permutations. That is, if X uses A; X uses B; or X uses both A and B, then "X uses A or B" is satisfied in any of the foregoing examples.

[0219] Furthermore, although this disclosure has been shown and described with respect to one or more implementations, equivalent variations and modifications will occur to those skilled in the art based on a reading and understanding of this specification and the accompanying drawings. This disclosure includes all such modifications and variations and is limited only by the scope of the appended claims. In particular, with respect to the various functions performed by the aforementioned components (e.g., elements, etc.), the terminology used to describe such components is intended to correspond to any component (unless otherwise indicated) that performs the specified function of said component (e.g., is functionally equivalent to it), even if structurally not equivalent to the disclosed structure performing the functions in the exemplary implementations of this disclosure shown herein. Moreover, although specific features of this disclosure have been disclosed with respect to only one of several implementations, such features may be combined with one or more features of other implementations that may be desirable and advantageous for a given or particular application. Furthermore, with regard to the use of the terms “comprising,” “having,” “containing,” or variations thereof in the Detailed Description or claims, such terms are intended to be included in a manner similar to the term “including.”

[0220] The functional units in this invention embodiment can be integrated into a processing module, or each unit can exist physically separately, or multiple units can be integrated into a module. The integrated module can be implemented in hardware or as a software functional module. If the integrated module is implemented as a software functional module and sold or used as an independent product, it can also be stored in a computer-readable storage medium. The storage medium mentioned above can be a read-only memory, a disk, or an optical disk, etc. The aforementioned devices or systems can execute the storage methods in the corresponding method embodiments.

[0221] In summary, the above embodiments are one implementation of the present invention, but the implementation of the present invention is not limited to the embodiments described above. Any changes, modifications, substitutions, combinations, or simplifications made that deviate from the spirit and principle of the present invention should be considered equivalent substitutions and are included within the protection scope of the present invention.

Claims

1. A method for calculating the trajectory of an object dropped from the air into water, characterized in that, Includes the following steps: Calculate the drag force F in the direction y of the water flow. D The upward force f in the direction of water depth z D : ； ； Calculate the effective gravity P: ； In the formula, C D C represents the drag coefficient in the direction of seawater flow, i.e., the drag force coefficient. L The drag coefficient in the downward direction is the coefficient of upward force; m is the mass of the bomb, m = p0 V; g is the acceleration due to gravity; p2 The density of water; p0 U is the density of the bomb; u0 is the velocity of the water flow around the bomb; u represents the velocity of the object in the direction of the water flow; A represents the velocity of an object sinking in the direction of its depth in water. h A is the equivalent area of ​​the horizontal cross section; P d is the equivalent area of ​​the cross section in the numerical direction; V is the volume of the object, and d is the diameter of the bomb. Establish the equation of state for the bomb falling through the air until it just touches the water surface: ； in , , ,have Integrating over time t, we get the equation: ； The state of the bomb when it enters the water: Considering the initial velocity in the z-direction when entering the water, we get: ； ； Will Substituting into the above equation, we have Substituting back into the above formula, according to: ， ， ， ； Perform an equation transformation, let ; Integrating over time t and rearranging, we obtain the system of equations: ； Substitute u0 and A h We obtain the time t and the horizontal displacement y of the bomb in the seawater; The horizontal displacement of the bomb in the air can be calculated using the following formula: ； Among them, v 0y Let be the initial velocity of the bomb in the y-direction in the air. Summing the horizontal displacements in the air and sea, we obtain the relative horizontal displacements between the aircraft and the submarine.

2. A method for calculating the trajectory of an object dropped from the air into water, characterized in that, Including the following steps: According to Newton's second law, the bomb satisfies the following differential equation: ； v y Let u0 be the absolute velocity in the vertical direction in the water, and c be the velocity of the water flow around the bomb. D denoted as , where is the drag coefficient in the direction of seawater flow, i.e., the drag force coefficient; m is the mass of the bomb. p2 Let u be the density of water; u0 be the velocity of the water flow around the bomb. The corresponding initial conditions are: when t=0, y=0, dy / dt=v y When = 0, the equation is: ； Integrating both sides yields ； Where a1 = A is the integration constant. Substituting the initial condition t=0, dy / dt = v y = 0, solving for 0 yields ； ； ； Finally, integrating both sides with respect to t, we get: ； Given the initial condition t=0, y=0, we can solve for B=0. Therefore: ； From the moment the bomb touches the water surface until it is just fully submerged, the bomb experiences buoyancy, water resistance, and gravity along the z-axis. The buoyancy is: ； c is the drag coefficient; The effect of gravity on the speed of a bomb is as follows: ； Where g is the acceleration due to gravity; The effect of buoyancy on bomb speed: ； In the formula S z is the bomb area along the Z-axis, and C and D are integration constants; Taking into account the effects of gravity, buoyancy, and water resistance, the following equation is obtained through velocity composition: ； ； ； During the stage from when the bomb is just fully submerged in the water to when it sinks to the bottom, ； Substituting the initial condition at t=0, z=-c / 2, we get , In summary, the parametric equations for the bomb's motion throughout the entire descent process are: ； Where d is the diameter of the submarine. ， ， ， ； Substituting the corresponding parameter values ​​into the above model, the horizontal displacement of the bomb in the seawater is obtained. The horizontal velocity and displacement of the bomb in the air are calculated. The horizontal displacements in the air and in the sea are summed to obtain the horizontal relative displacement between the aircraft and the submarine and the vertical displacement between the aircraft and the submarine.

3. The method for calculating the trajectory of an object dropped from the air into water according to claim 2, characterized in that, When the target submarine's possible diving depth is within a certain range, the following model is established: ； At 6 seconds, the speed drops to 0 and begins to rise. When the submarine sinks to a depth of 20m to 50m, the bomb is effective; when the submarine sinks to a depth of more than 50m, the bomb has no power and is only subject to buoyancy, and is ineffective against the submarine. 1) The aircraft's flight direction is the same as the submarine's direction of travel. The optimal time to drop bombs is when the submarine's coordinates relative to the aircraft are (x1, y1+0.5, -20) to (x1, y1+5, -50). The optimal bombing area for the aircraft is (x1, y1 -59.343, -20) to (x1, y1+42.76, -50). 2) The aircraft's flight direction is opposite to the submarine's direction of travel. The optimal time to drop bombs is when the submarine's coordinates relative to the aircraft are (x1, y1+0.5, -20) to (x1, y1+5, -50). The optimal bombing area for the aircraft is (x1, y1 +40.657, -20) to (x1, y1-22.76, -50). 3) The aircraft's flight direction is perpendicular to the ship's direction of travel. The optimal time to drop bombs is when the ship's coordinates relative to the aircraft are between (x1+1.8, y1, -30) and (x1+2.5, y1, -30). The optimal bombing zone for the aircraft is between (x1-29.65, y1, -30) and (x1+60.8, y1, -30). 4) When the aircraft's flight direction is at a 45° angle to the ship's direction of travel, the optimal time to drop bombs is when the ship's coordinates relative to the aircraft are (x1+15, y1+5, -18) to (x1+15, y1+5, -18); the optimal bombing zone for the aircraft is (x1-30, y1, -30) to (x1+18, y1, -30).

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