Abstract
The motion of the object in the medium has always been a hot research topic, and it is closely connected with many applications in our life. The acceleration of the object with multiple forces becomes very complicated, especially when these forces depend on the motion of the object. The exact formula for the object motion is a differential-integral equation and is very difficult to be solved analytically. One example of this kind of motions is the rocket launch. With sufficient thrust, the rocket can obtain an acceleration large enough to escape from the gravity of the earth. With the increasing height, the gravity from the earth becomes smaller, which affects the net acceleration of the rocket. Meanwhile, the air resistance becomes more and more important when the velocity of the rocket increases. It even plays the main role in the middle stage of the launch. Also, as the air resistance depends on both the velocity of the rocket and the air density (there is no air resistance in vacuum), the air resistance will decrease when the air density becomes small enough at the large height. In this article, a model that includes all of the factors mentioned above is established, and how these forces change the velocity of the rocket is analyzed. Two scenarios, one with air resistance and one without, are described. The velocity of the rocket in each scenario is represented by graphs, which are compared. With justification, the Taylor series is used to solve the differential-integral equation, and it is found that the fuel thrust and the gravity become important in the rocket launch at the beginning stage. In the middle stage, the air resistance begins to have a significant effect and reduces the acceleration of the rocket. In the final stage, there is virtually no gravity or air resistance, and only the fuel thrust contributes to the acceleration of the rocket.
Highlights
The acceleration process is very common in nature
The final velocity of the rocket is determined by the fuel thrust and the height dependence of the gravity constant which is determined by the formula below
This paper proposes a theoretical model with differential-integral equation to simulate the rocket launch in the air by including the fuel thrust, gravity, and the air resistance
Summary
The acceleration process is very common in nature. It ranges from the acceleration of racing cars and the flow of water on mountains, to the launch of rockets. This paper introduces many factors, including gravitational acceleration, air resistance, as well as the motion of a variable mass object to solve the problem of the acceleration of a rocket [1,2,3]. As the vehicle speed increases, and the atmosphere thins, there is a point of maximum aerodynamic drag called Max Q This determines the minimum aerodynamic strength of the vehicle, as the rocket must avoid buckling under these forces.[10]. Modern rockets use the method of gradually ejecting the burned gas outward to increase the speed of the rocket itself This model obviously belongs to the problem of a variable-mass system. Many factors mentioned above are considered, and the results are analyzed
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