Analytical description of ascending motion of rockets in the atmosphere

Rodrigues, H.; Pinho, Marcos; Portes, D.; Santiago, A. J.

doi:10.1088/0143-0807/30/1/019

Cited by 4 publications

(5 citation statements)

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“…Naturally, when analysing launches from Earth one must include the effects of gravity and air drag, which are largest on the first stage. Reference [1]'s appendix extends the rocket equation to vertical motion in a gravitational field, while [11] analyses ascent through an atmosphere. Often, these effects are included in aggregate by assigning ∆v grav and ∆v drag 'penalties' for an Earthlaunched rocket to overcome before it reaches orbit.…”

Section: Discussionmentioning

confidence: 99%

Learning about rockets, in stages

Blanco¹

2022

Phys. Educ.

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A rocket must carry the fuel it expels in order to accelerate its structure and payload. The rocket equation relates the change in speed to the fuel mass expelled. To launch a spacecraft into Earth orbit requires a multi-stage rocket, since otherwise the mass of fuel required would be prohibitive. While the details vary among historical and current launch vehicles, the advantages and physical principles of staging can be demonstrated by analysing the simpler case of a two-stage rocket in free space. I show that a two-stage rocket produces a higher payload speed than a single-stage rocket with the same fuel and structural mass, and introduce the concept of optimal staging to maximise this speed. I also examine how mechanical energy extracted from the expelled fuel is distributed between the stages, payload capsule, and exhaust. A surprising result is that the two-stage rocket deposits more energy into the exhaust stream than its single-stage counterpart. I suggest further investigations that are appropriate for advanced high school or undergraduate students of physics or aerospace engineering. Supplementary materials are provided online for classroom use.

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Section: Discussionmentioning

confidence: 99%

Learning about rockets, in stages

Blanco¹

2022

Phys. Educ.

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“…Neste trabalho, utilizou-se a programação em Python [22] para a obtenção de tabelas de dados. Posteriormente utilizou-se o programa Origin [23] para gerar os gráficos com os dados das tabelas.…”

Section: Método Numérico De Eulerunclassified

“…O valor médio calculado para o impulso total foi: I méd tot = (35 ± 1)N.s. Adicionalmente, com os valores de m p e t q foi possível estimar, utilizando a Equação (23), o parâmetro dm/dt. A grandeza t q foi obtida com o altímetro acoplado ao Epsilon-8, durante seu voo, conforme se discute na próxima seção.…”

Section: Estimativas Teóricas/experimentais Extraídas Do Motor E Do V...unclassified

Soluções analítica e numérica para o movimento vertical de um minifoguete

Alves,

Bento,

Marchi

2023

Rev. Bras. Ensino Fís.

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Neste trabalho é descrito um modelo para a solução analítica da equação diferencial de movimento de um minifoguete que descreve uma trajetória vertical num movimento de subida. As soluções para a velocidade, v(t), e altitude, y(t), são comparadas com uma solução numérica utilizando o método de Euler. Para testar estes modelos e comparar os métodos para a obtenção de v(t) e y(t), foi utilizado o voo experimental do minifoguete Epsilon-8 com motor a propelente sólido, do grupo de foguetemodelismo Carl Sagan da Universidade Federal do Paraná (UFPR). Na solução analítica é considerado um modelo aproximado no qual a massa total do minifoguete é constante durante o tempo de queima do propelente. As previsões numérica e analítica para v(t) e y(t) podem ser obtidas utilizando parâmetros físicos relacionados ao minifoguete: as curvas experimentais para o empuxo, T(t), e a força de arrasto. Foram utilizadas curvas T(t) de quatro motores fabricados pela empresa Boa Vista Modelismo e testados pelo grupo de Carl Sagan, que são do mesmo lote do motor usado no voo experimental do Epsilon-8. Embora o método analítico seja satisfatório, o método numérico retorna valores de velocidade e altitudes mais próximos ao voo experimental do Epsilon-8.

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“…However, it is necessary to know the dependence of the mass on the time if we want to express the velocity as a function of time. In this sense, in the literature the constancy of the mass exhaustion rate is widely assumed [13]. In contrast, the purpose of this work is to study the motion of a self-propelled body assuming that the mass exhaustion rate is a decreasing exponential function of time.…”

Section: Exponential Mass Exhaustionmentioning

confidence: 99%

Modelling the dynamics of bodies self-propelled by exponential mass exhaustion

Rodrigues¹,

Pinho²,

Portes³

et al. 2008

Eur. J. Phys.

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We present a study of the ascending vertical motion of a self-propelled body under a uniform gravitational field suffering the action of two different types of air friction forces: linear on the velocity, which is valid for slowly moving bodies, and quadratic on the velocity. We study the special case where the thrust force is a decreasing function of mass, which corresponds to the exponential mass exhaustion rate. We present in detail the analytical solutions for the equations of motion for the two types of air friction, and briefly present some techniques for solving the related ordinary differential equations. This paper is intended for undergraduate physics teachers and for graduate students.

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Analytical description of ascending motion of rockets in the atmosphere

Cited by 4 publications

References 1 publication

Learning about rockets, in stages

Learning about rockets, in stages

Soluções analítica e numérica para o movimento vertical de um minifoguete

Modelling the dynamics of bodies self-propelled by exponential mass exhaustion

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