Tsiolkovsky Rocket Equation Improvements

Applied Sciences

This papers introduces an analytic method to define multistage launcher trajectories to determine the payload mass that can be inserted in orbits of different semimajor axes and inclinations. This method can evaluate the gravity loss, which is the main term to be subtracted to the Tziolkowski evaluation of the velocity provided by the thrust of a launcher. In the method, the trajectories are dependent on two parameters only: the final flight-path angle γf at the end of the gravity-turn arc of the launcher trajectory and the duration tc of the coasting arc following the gravity-turn phase. The analytic formulas for the gravity-turn phase, being solutions of differential equations with a singularity, allow us to identify the trajectory with a required final flight-path angle γf in infinite solutions with the same initial vertical launch condition. This can also drive the selection of the parameters of the pitch manoeuvre needed to turn the launcher from the initial vertical arc. For a...

2012

In this paper, we consider a nonlinear second order equation modelling rocket motion in the gravitational field obstructed by the drag force. The proofs of the main results are based on topological fixed point approach. 1991 Mathematics Subject Classification. 34B10, 34B15, 35A15, 35J25, 93C15.

Propulsion and Power Research, 2013

In this study, applications of some analytical methods on nonlinear equation of the launching of a rocket with variable mass are investigated. Differential transformation method (DTM), homotopy perturbation method (HPM) and least square method (LSM) were applied and their results are compared with numerical solution. An excellent agreement with analytical methods and numerical ones is observed in the results and this reveals that analytical methods are effective and convenient. Also a parametric study is performed here which includes the effect of exhaust velocity (C e), burn rate (BR) of fuel and diameter of cylindrical rocket (d) on the motion of a sample rocket, and contours for showing the sensitivity of these parameters are plotted. The main results indicate that the rocket velocity and altitude are increased with increasing the C e and BR and decreased with increasing the rocket diameter and drag coefficient.

Journal of Mechanical Engineering and Technology, 2013

In this study, the mathematical investigation of rocket motion path is presented using analytical and numerical methods. Differential transform method and fourth-order Runge-Kutta method are used for solving the current problem. A parametric study is performed which includes the effect of exhaust velocity (C e), burn rate (BR) of fuel and diameter of cylindrical rocket (d) on the motion of a sample rocket. Also a realistic comparison of four on-line and active rockets is presented and the result is discussed. The main results indicate that the present analytical technique has high accuracy in comparison with the numerical solution. It is illustrated that the rocket velocity and altitude are increased with increasing the C e and BR and decreased with increasing the rocket diameter. Finally it is concluded that Mu-3S-II rocket reaches to higher velocities and altitudes faster in comparison with the other introduced rockets.

To achieve an accurate rocket launch, it is necessary to assess how the perturbations appearing when the rocket leaves the rocket launcher system, influence the rocket flight trajectory and shooting accuracy as well. Therefore, we will analyse the influence of the rocket launcher system oscillations, during the shooting, on the rocket movement. We are taking into account other perturbation factors that can appear during the shooting and the rocket flight in standard atmosphere.

EDULEARN proceedings, 2017

The achievement of a comprehensive education of the students to provide a global knowledge that allows them to make appropriate decisions in the framework of their future work is one of the most important challenges of the Engineering degrees, especially in basic subjects as Physics or Mathematics. In this sense, a great effort has to be done by the teachers of the first year where the subjects are generally quite abstract and, as a consequence, the motivation of the students is hard to attain. Several are the conditions to obtain this goal successfully: First, a correct cross-coordination and a well-designed teaching of the basis subjects to avoid undesirable repetition of some concepts under different point of view and to show the basic knowledge as a uniform conceptual body ready to be applied to real situations. Second, the need to make attractive the study of these basic and abstract subjects by proposing the resolution of real problems related with the goals of each particular Degree, and adapted to the knowledge of the student of the first level. These problems have to be solved involving different branch of the basic subjects as Physics or Mathematics. In this paper we present a real project related to the professional skills of the Geomatic and Topography Degree, adapted for the first level, and involving three core subjects namely, Algebra, Calculus and Mechanics. The approach of this real project, proposed as an interdisciplinary practice, is related with the path followed by a rocket that puts a satellite in orbit. In real situations, sixteen are the steps of the protocol to achieve this goal, from the takeoff of the rocket to its entrance on the final elliptic orbit. To adapt this complicated protocol to our purposes, we have simplified the steps considering finally only four: (i) the takeoff of the rocket; (ii) the change of the upright path at a predetermined height from the ground to approach to the elliptical orbit; (iii) the link between the rise of the rocket and its placing in elliptical orbit; (iv) the putting into the elliptical orbit itself. Analysis of classic geometric figures as the conics, with their different forms of representation, becomes necessary to be able to develop this practice, since they adapt to the forms described by orbits of the satellites. Modeling makes it necessary to use different reference systems and their relationship between them. In addition, the simplified study of the satellite position vector in its different phases provides fundamental data such as speed, acceleration or balance of forces, related with the subjects of Dynamic or Kinematics, and where the concept of derivative plays a fundamental role. A later detailed study on the motion of the satellite would lead to propose systems of differential equations that relate the elements studied and whose complexity requires numerical methods for their resolution.

Cuadernos de Ingeniería Matemática, 2021

Mathematics

The projectile motion (PP) in a vacuum is re-examined in this paper, taking into account the relativistic mass in special relativity (SR). In the literature, the mass of the projectile was considered as a constant during motion. However, the mass of a projectile varies with velocity according to Einstein’s famous equation m=m01−v2/c2, where m0 is the rest mass of the projectile and c is the speed of light. The governing system consists of two-coupled nonlinear ordinary differential equations (NODEs) with prescribed initial conditions. An analytical approach is suggested to treat the current model. Explicit formulas are determined for the main characteristics of the relativistic projectile (RP) such as time of flight, time of maximum height, range, maximum height, and the trajectory. The relativistic results reduce to the corresponding ones of the non-relativistic projectile (NRP) in Newtonian mechanics, when the initial velocity is not comparable to c. It is revealed that the mass o...

We consider the orbit of projectiles launched with arbitrary speeds from the Earth's surface. This is a generalization of Newton's discussion about the transition from parabolic to circular orbits, when the launch speed approaches the value ν = √ gR E . We find the range for arbitrary launch speeds and angles, and calculate the eccentricity of the elliptical orbits.

Journal of Optimization Theory and Applications, 2012

We derive the covariant optimality conditions for rocket trajectories in general relativity, with and without a bound on the magnitude of the proper acceleration. The resulting theory is then applied to solve two specific problems: the minimum fuel consumption transfer between two galaxies in a FLRW model, and between two stable circular orbits in the Schwarzschild spacetime.