Realistic Simulations of Non-Linear Acceleration of the Rocket in the Air

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.