{\left( {\ln m} \right)} \right|_{{m_0}}^{{m_1}},\;\;}\Rightarrow

Differential transformation method (DTM), homotopy perturbation method (HPM) and least square method (LSM) were applied and their results are compared with numerical solution. As a result, the equation is written as follows:\[

Although there are many cases for which this particular model is

In this study, applications of some analytical methods on nonlinear equation of the launching of a rocket with variable mass are investigated.

The resulting formula is called the To get this formula it’s convenient to use the differential equation in the form:Take into account that \(dm\) denotes mass decrease. Rocket motion is based on Newton’s third law, which states that “for every action there is an equal and opposite reaction”. By continuing you agree to the Copyright © 2020 Elsevier B.V. or its licensors or contributors. Such kind of motion often occurs in nature and technology. v \right|_{{v_0}}^{{v_1}} = – u\left. 2.1. The behavior of a rocket in flight has been studied extensively. Rocket flight. The more gas is ejected from the rocket ,the mass of the rocket decreases; To analyse this process let us consider a rocket being fired in upwards direction and we neglect the resistance offered by the air to the motion of the rocket and variation in the value of the acceleration due to gravity with height \]where \({v_0}\) and \({v_1}\) are the initial and final velocities of the rocket, \({m_0}\) and \({m_1}\) are the initial and final masses of the rocket, respectively.By setting \({v_0} = 0,\) we obtain the formula derived by Tsiolkovsky:This formula determines the rocket velocity depending on its mass change while the fuel is burning. In the given section we consider moving bodies with variable mass. It can be confusing to try to apply Newton's second law of motion directly to such a system. {{v_1} – {v_0} = u\ln \frac{{{m_0}}}{{{m_1}}}.}

It allows rough estimation of the fuel capacity necessary to accelerate the rocket to a given velocity.This website uses cookies to improve your experience.

= (9.27) However, this equation is correct in only a few very special cases. An excellent agreement with analytical methods and numerical ones is observed in the results and this reveals that analytical methods are effective and convenient.
Production and hosting by Elsevier B.V.ScienceDirect ® is a registered trademark of Elsevier B.V. Differential Equation of Rocket Motion. Newton’s Second Law for Systems with Variable Mass David Chandler, Porterville College, 100 E. College Ave., Porterville, CA 93257; [email protected] T he usual model used to illus-trate variable-mass problems is a rocket. This leads the rocket velocity to be increased by \(dv.\) We apply the \[{{p_1} }={ \left( {m – dm} \right)\left( {v + dv} \right),}\]and the momentum of the exhaust gases in the Earth’s coordinate system isBy the law of conservation of the total momentum of the system, we can write:We can neglect the term \(dmdv\) in the last expression considering small increments of these values. {\left. We now wish to discuss the dynamics of a system, such as a rocket, whose mass varies. In this study, applications of some analytical methods on nonlinear equation of the launching of a rocket with variable mass are investigated.

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