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Modeling Rocket Flight in the Low-Friction Approximation Undergraduate Journal of Mathematical Modeling: One + Two Volume 6 | 2014 Fall Article 5 2014 Modeling Rocket Flight in the Low-Friction Approximation Logan White University of South Florida Advisors: Manoug Manougian, Mathematics and Statistics Razvan Teodorescu, Physics Problem Suggested By: Razvan Teodorescu Follow this and additional works at: https://scholarcommons.usf.edu/ujmm Part of the Mathematics Commons UJMM is an open access journal, free to authors and readers, and relies on your support: Donate Now Recommended Citation White, Logan (2014) "Modeling Rocket Flight in the Low-Friction Approximation," Undergraduate Journal of Mathematical Modeling: One + Two: Vol. 6: Iss. 1, Article 5. DOI: http://dx.doi.org/10.5038/2326-3652.6.1.4861 Available at: https://scholarcommons.usf.edu/ujmm/vol6/iss1/5 Modeling Rocket Flight in the Low-Friction Approximation Abstract In a realistic model for rocket dynamics, in the presence of atmospheric drag and altitude-dependent gravity, the exact kinematic equation cannot be integrated in closed form; even when neglecting friction, the exact solution is a combination of elliptic functions of Jacobi type, which are not easy to use in a computational sense. This project provides a precise analysis of the various terms in the full equation (such as gravity, drag, and exhaust momentum), and the numerical ranges for which various approximations are accurate to within 1%. The analysis leads to optimal approximations expressed through elementary functions, which can be implemented for efficient flight prediction on simple computational devices, such as smartphone applications. Keywords Differential Equations, Rocket Flight, Motion This article is available in Undergraduate Journal of Mathematical Modeling: One + Two: https://scholarcommons.usf.edu/ujmm/vol6/iss1/5 White: Modeling Rocket Flight in the Low-Friction Approximation MODELING ROCKET FLIGHT IN THE LOW-FRICTION APPROXIMATION 3 PROBLEM STATEMENT & MOTIVATION The question under investigation in this paper is: How can we best model rocket flight with closed-form equations? MATHEMATICAL DESCRIPTION AND SOLUTION APPROACH I. EXACT SOLUTIONS Assuming that the relationship between the mass of the rocket 푚(푡) at time 푡 and the rate of mass depletion 푚′(푡) is proportional, gives 푑푚 = −푄 푚(푡) 푑푡 for some constant 푄. Hence the mass remaining at time 푡, found through separation of variables and subsequent integration, is −푄 푡 푚(푡) = 푚0 푒 where 푡 ≥ 0 (1) and 푚0 = 푚(0) is the initial mass of the rocket. By Newton’s Second Law, the sum of forces 퐹 on an object equals the product of the object’s mass and acceleration: 푛 푑푣 ∑ 퐹 = 푚 . (2) 푑푡 =1 The forces summed in the direction of the rocket’s flight following liftoff are the gravitational force: Produced by The Berkeley Electronic Press, 2014 Undergraduate Journal of Mathematical Modeling: One + Two, Vol. 6, Iss. 1 [2014], Art. 5 4 LOGAN WHITE 퐺 푚 푚(푡) 퐹 = − 푒 , (3) 푦2 where G is Newton’s gravitational constant, 푚푒 is Earth’s mass, and 푦 = 푦(푡) is the vertical position of the rocket relative to Earth’s center, and the force of thrust caused by the ejection of fuel out of the rocket’s nozzle: 푑푚 퐹 = −푐 , (4) 푡 푑푡 where 푐 is the constant exhaust speed, relative to the rocket. Letting 훼 = 퐺 푚푒 and 훽 = 푄 푐 and combining equations (2), (3), and (4), we receive a differential equation into which we can substitute equation (1) like so: 푑푣 훼 푚(푡) 푑푚 푑푣 훼 푚 푒−푄푡 푚(푡) = − − 푐 ⟹ (푚 푒−푄푡) = − 0 + 훽 (푚 푒−푄푡) . 푑푡 푦2 푑푡 0 푑푡 푦2 0 Simplifying, we receive 푑푣 훼 = 훽 − , (5) 푑푡 푦2 Noticing that 푣 = 푑푦/푑푡 : 푑푣 푑푣 푑푦 푑푣 = = 푣 . (6) 푑푡 푑푦 푑푡 푑푦 Substituting equation (6) into our second-order differential equation (5), we arrive at the equation: 푑푣 훼 훼 푣 = 훽 − ⟹ 푣 푑푣 = 훽 푑푦 − 푑푦 푑푦 푦2 푦2 and integrate: https://scholarcommons.usf.edu/ujmm/vol6/iss1/5 DOI: http://dx.doi.org/10.5038/2326-3652.6.1.4861 White: Modeling Rocket Flight in the Low-Friction Approximation MODELING ROCKET FLIGHT IN THE LOW-FRICTION APPROXIMATION 5 푣(푡) 푦(푡) 푦(푡) 훼 [ 푣(푡) ]2 훼 [ 푦(푡) − 푟 ] ∫ 푣 푑푣 = ∫ 훽 푑푦 − ∫ 푑푦 ⟹ = 훽 [ 푦(푡) − 푟 ] − 푒 , 푦2 2 푒 푟 푦(푡) 푣=0 푦=푟푒 푦=푟푒 푒 noting that the rocket begins its flight on the Earth’s surface, i.e., 푦(0) = 푟푒 where 푟푒 is the radius of the Earth. If we substitute 푣(푡) = 푑푦/푑푡, we arrive at 2 푑푦 훼 [푦 − 푟푒] ( ) = 2 훽 [푦 − 푟푒] − 2 . (7) 푑푡 푟푒 푦 Separating variables in equation (7) gives 푑푦 푑푡 = 훼 [푦 − 푟푒] √2 훽 [푦 − 푟푒] − 2 푟푒 푦 and integrating yields 푡 푦(푡) 푑푦 푧(푡) 푑푧 ∫ 푑 푡 = ∫ = ∫ , 푡=0 푦=푟푒 훼 푧=0 1 (8) √2[푦 − 푟푒] [훽 − ] − 푟푒푦 2훼 푟푒훽 푟푒 √ [푧] [ + 푧 ] 푟푒 훼 1 − (− ) 푟푒 and we get an equation that is difficult to use. This solution is not practical from a computational standpoint, as it involves two different types of Jacobi elliptic integrals. Instead, it would probably be more useful to investigate various methods of approximation by which we can simplify the function further. 1. APPROXIMATIONS AND ERROR 1 − 푟푒 The simplest approximation of equation (8) that we consider neglects the effect of the 푧(푡) 1−(− ) 푟푒 term. This approximation leads to a solution that is unsatisfactory because it ignores the effects of gravity: Produced by The Berkeley Electronic Press, 2014 Undergraduate Journal of Mathematical Modeling: One + Two, Vol. 6, Iss. 1 [2014], Art. 5 6 LOGAN WHITE 훽 푧(푡) = 푡2 . (9) 2 In order to arrive at a more satisfactory solution, we will have to look at geometric series. The 푎 sum of the infinite and convergent geometric series ∑∞ 푎 is 1 , where 푎 is the first term of =1 1−푟 1 the series and 푟 is the common ratio. The term in the denominator of the right side of equation (8) that makes the equation difficult to integrate is 1 − 푟 1 푧(푡) [푧(푡)]2 푒 = − + − + ⋯ . (10) −푧(푡) 푟 푟2 3 1 − ( ) 푒 푒 푟푒 푟푒 1 푧(푡) when expressed as the sum of an infinite geometric series with 푎1 = − and 푟 = − 푟푒 푟푒 The total sum (10) can be approximated by simply taking the first few terms of the series. All of this can be done only under the assumption that 푧(푡) ≪ 푟푒, which makes the series convergent. However, this is always going to be true for the first stage of any multi-stage rocket flight. The error of any order approximation is going to be less than 0.01 (=1%) as follows: 1 푧(푡) [푧(푡)]2 푆푛 = − + 2 − 3 + ⋯ + 푎푛 + ⋯ . 푟푒 푟푒 푟푒 푎푛 is neglected as having less than 1% of the total sum 푆푛. So 1 푧(푡) [푧(푡)]2 푆푛 = − + 2 − 3 + ⋯ + 푅푛, 푟푒 푟푒 푟푒 where 푅푛 is the remainder and error term used to represent the terms 푎푛 and beyond. In particular, https://scholarcommons.usf.edu/ujmm/vol6/iss1/5 DOI: http://dx.doi.org/10.5038/2326-3652.6.1.4861 White: Modeling Rocket Flight in the Low-Friction Approximation MODELING ROCKET FLIGHT IN THE LOW-FRICTION APPROXIMATION 7 푎 푅 = 푛 . 푛 푧(푡) 1 − (− ) 푟푒 Because 푎푛 < 0.01, it must be true that 푅푛 < 0.01, so all errors are less than 1%. i. ZERO-ORDER APPROXIMATION In the zero-order approximation, the first term is taken from the infinite series (10) and used to approximate the sum of the series. When this substitution is carried out and the integrand simplified, we obtain 푧(푡) 푑푧 1 푡 = ∫ = √2 푧(푡) (훽 − 푔) . 푧=0 √2푧(푡)(훽 − 푔) 2(훽 − 푔) Solving for 푧(푡), we get 푡2 푧(푡) = (훽 − 푔). (11) 2 푧(푡) If the zero-order approximation is only reasonable when ≤ 0.01, the time domain 0 ≤ 푡 ≤ 푇0 푟푒 can be found as follows: 훽 − 푔 0.02 푟 0.01 푟 = 푇2 ( ) ⇒ 푇 = √ 푒 . 푒 0 2 0 훽 − 푔 Equation (11) provides a quadratic approximation of the first stage of a rocket flight; however, it does not account for variation in gravitational force. Instead, it assumes a constant force 푚푔 that would only be present at Earth’s surface. Because we are assuming a varying gravitational force, it is necessary to use a first-order approximation. ii. FIRST-ORDER APPROXIMATION Produced by The Berkeley Electronic Press, 2014 Undergraduate Journal of Mathematical Modeling: One + Two, Vol. 6, Iss. 1 [2014], Art. 5 8 LOGAN WHITE Taking the first two terms of the series (10) and substituting in for the sum of the series, we arrive at 푡 푧(푡) 푑푧 푧(푡) 푑푧 ∫ 푑푡 = ∫ = ∫ 2 2 푡=0 푧=0 푧=0 √퐴 [(푧 + 휅) − 휅 ] (12) 2훼 푟푒 훽 1 푧 √ (푧) ( − + 2) 푟푒 훼 푟푒 푟푒 2 훽− where 퐴 = and 휅 = . Using the trigonometric substitution 푧 = 휅 sec 휃 − 휅 and 푑푧 = 푟푒 퐴 휅 sec 휃 tan 휃 푑휃, equation (12) evaluates to 푡 1 휅 sec 휃−휅 휅 sec 휃 tan 휃 ∫ 푑푡 = ∫ 푑휃 2 2 2 푡=0 √퐴 휃0 √휅 sec 휃 − 휅 휅 sec 휃−휅 = ∫ sec 휃 푑휃 휃0 푧(푡) 1 푧 + 휅 + √(푧 + 휅)2 − 휅2 = ln | | . √퐴 휅 푧=0 Evaluated at its upper and lower limits, this equation gives us our position function of time, 푧(푡), which is a hyperbolic cosine function: 푧(푡) = 휅 cosh(푡√퐴) − 휅 .
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