Math Exploration:Trajectory of a RocketBy Mridvika SureshWord Count : Background and Rationale In TV shows or movies, when they show a rocket being launched, it always seems that the rocket is directly launched upwards and travels in a projectile or curved path until it reaches its destination. At least this is the thought I always seemed to believe in. Last week when I found out the truth about rockets, I had to know more about how they worked. The rocket travels in a straight direction (upwards) till it reaches the outer layer of earth and to leave the planet to travel to another one, it travels in a spiral motion. This is so that the rocket experiences less aerodynamic stress. After reaching the atmosphere, the rocket stops using its fuel and uses the earth’s gravity to change its direction for the trans-lunar injection. Introduction The aim of my exploration is to model the trajectory of a rocket. I will be considering different rockets from the Apollo missions and research on the different ways to find the trajectory and the factors affecting the trajectory of the rocket. To do this, I will be calculating the equations of the elliptical orbits, use polar coordinate method to get the different trajectories and will find the equations of the elliptical orbits to look for a comparison between the 2 factors. I will be doing this with the help of differential equations, Tsiolkovsky rocket equation and equation of ellipses. Tsiolkovsky rocket equation is used to determine the speed of a rocket. I will need to find the speed of the rocket as there is a relation between the speeds the rocket is launched at and the elliptical path it travels in. PART 1 : Using CalculusFor this part, I’ll be using a method called a ‘polar coordinate’. This method involves 2 equations which help model the flight of the rocket. The first part involves finding the velocity, and the second part will involve calculating the angle. Since the path is a curved one, a tangent is taken and the equations for velocity and angle are combined and solved using the Runge-Kutta method. 1: Modelling velocity From Newton’s second law; Newton’s second law states F=ma (force= mass x acceleration). In this case, to calculate final forces acting on the rocket, using vectors, it is shown that:Force = thrust – drag – weightThere is another equation for drag which can be used to replace the ‘d’ value in the equation above. The equation for drag is D = 0.5 C ? A V2 C – “coefficient of drag” ? – air density A – the cross sectional area of the rocket v – velocity. As weight acts vertically down, we must calculate the component of weight which acts parallel to the rocket’s velocity, which is given by the equation weight = mgsin This gives our first equation (where T is equal to thrust).dvdt= T – (12 Cd p A v2) – (mgsin)m(ENTER EQUATION HERE)2: Modelling rotation We use circular motion to model the rocket’s rotation. The force acting perpendicular to the rocket’s velocity is known as the centripetal force, and is equal to the component of weight acting in that direction. Since we know that; ( ) ( ) This can be substituted into the previous equation to give; Then simplified and rearranged for our second equation. Note that ? is negative as the centripetal force will reduce the angle of the rocket. This function will be referred to as ( ) The 2 rockets I’ll be using for comparison is the first and the last successful mission to the moon:APOLLO 11APOLLO 16MASS28,801 kg210 kgAREA65.8 mPatched Conic Technique is another way to….The ?V using instantaneous (e.g. chemical propulsion) maneuvers can be determined by repeated application of this equation that simply says that the total energy is the sum of the kinetic energy and the potential energy:E =v22-r In this equation the total energy per mass or specific energy is denoted by E, velocity is v, is the GM of the central body and r is the distance from the center of the body. The key is that the total energy of the object is a constant of motion over the orbit.We will also use the fact that orbits are ellipses, and this equation, which determines that constant of motion from the apses of the orbit, i.e. the radii of the closest and farthest points in the orbit, r1r1 and r2r2:LimitationsAlthough I covered most of the factors in calculating the trajectory, there were a few I had left out. Maximum Dynamic Pressure was one of the factors I didn’t consider. Denoted by max Q, maximum dynamic pressure