Rocket Motion

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Motion of the rocket is simulated using two numerical analysis methods. From the simulation different parameters such as altitude, velocity, acceleration and range for initial fuel flows were calculated. Two numerical methods, Euler’s integration and 4th order Runge-Kutta integration are used for calculating different parameters for the vertically launched rocket. The efficiency and the accuracy of the methods were compared. It was found out that the 4th order Runge-Kutta is more efficient than Euler’s integration method for the given time step. Also for the rocket given the optimal initial fuel mass flow rate for attaining the highest altitude is found to be 35.5 Kg/S which gives an altitude of 1362594 m. 1.0 INTRODUCTION

Rockets are important part of space travelling. But rockets are also in many other important applications. The basic understanding of the physics behind rocket motion is easier to understand as it obeys Newton’s laws of motion. But this understanding is not enough to design and test a rocket as there are other critical parameters that must be taken into account. It is critical to know the trends in the rocket parameters such as its velocity, distance travelled and acceleration in order to design rocket for its appropriate application. For this simulating the motion of the rocket and analysing the data measured is one of the efficient ways. In this report two numerical methods, Euler’s integration and 4th order Runge-Kutta integration are used for calculating different parameters for a vertically launched rocket. This report also discusses the trends in behaviour of some of the parameters measured and in particular the efficiency and accuracy of the both methods.


Figure x: Forces acting on Rocket
From Figure X and using Newton’s Laws of Motion the Net Force acting on the Rocket is given by the equation: Fnet = Thrust- Drag- Fgravity(1)
Fnet = Total Force on rocket
Fgravity = Force exerted on rocket by Earth or Weight of rocket(as indicated in Figure 1)
Thrust = Force from burning rocket fuel
Drag = Air resistance Force on rocket

Acceleration of Rocket = Force on rocket / Mass of Rocket
But the mass of the rocket changes with fuel flow rate. i.e. mass changes with time. So acceleration can only calculated by predicting the altitude (distance rocket travelled from launch to return) and velocity with respect to time. The density of atmosphere changes as the altitude increases. This means that the drag acting on the rocket is not constant and changes with the altitude as is the function of density and velocity. See equation (5) and (6) Density ρ, =1.2* e-y7000(5)

Drag Fd, =0.1*ρ*v*v(6)


For calculating the both methods Visual Basic Application (VBA) with Microsoft Excel was used. Codes for implementing Euler’s method and 4th order Runge-Kutta method was done in VBA and results were tabulated and graphs were plotted in Excel. See Appendix x for the Codes for Euler’ integration and 4th order Runge-Kutta. The initial values of the parameters were set to initialise the calculations; When time t =0,

Altitude y, = 0
Velocity v, = 0
Initial fuel flow rate Q0, = 20 Kg/S
Time step h, = 1 (this was changed depending on the method used) 4.0 RESULTS

4.1 Euler Method
The results below are obtained by using Euler’s integration method for the motion of rocket. Figure 2 shows the graph of maximum altitude against a range of time step, 0.01 to 1. From this graph the time step that is needed for the error in the maximum altitude to be approximately 0.1% was found to be 0.7, which is indicated by the straight horizontal line drawn in the graph. As the time step increases the...
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