Rocket Propulsion
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2D DesignTan Gee Yang, Dion Teo Jian Xian, Low Hui An Shawn, Albert Suryanto, Soh Jun XianTable of content:IntroductionPhysicsMath ChemistrySummaryAppendicesIntroductionThis report investigates on the effects of modifications on the nozzle/reducer on cable-guided rocket propulsion system (CGRPS). The report will focus on the theories and calculations of the CGRPS that will assist in the team to modify the most appropriate reducer in achieving a target distance of 17 to 18 m.By taking into consideration of the fixed chemical composition of the sugar fuel and studying the forces on this CGRPS, we will explore using various methods of increasing the mass of the nozzle/reducer and to modify the shape of it to increase air resistance so that the CGRPS will slow down sufficiently to reach within the target distance of 17 – 18 m.Physics:Dynamics of GPRSWhen the core is ignited, the combustion of fuel within the core of the rocket causes a propulsion of gaseous particles onto the air in the -x direction. By Newton’s 3rd Law of motion, this cause an equal and opposite thrust on the rocket in the +x direction. [pic 1]Initially, the rocket will stay at rest and only moves when the thrust is much bigger than the static friction between the rocket and the cable. According to Newton’s 2nd Law of motion, the rocket will accelerate towards the +x direction.[pic 2][pic 3][pic 4]When the rocket is in motion, it experiences aerodynamic friction and kinetic friction which act in the opposite direction to the thrust. Both aerodynamic friction varies with the velocity of rocket while kinetic friction varies with the total mass of the rocket, where aerodynamic friction:  and kinetic friction: .[pic 5][pic 6]As a result, the resultant force acting on the rocket can be described in the equation below:=.[pic 7][pic 8]Our Team’s StrategyBased on our observation during the firing test, our group realized that the rocket exceeds the target distance, which is between 16m and 17m. Thus, we decided to add 10 marbles and a wooden board in order to increase the frictional forces. Our modification increased the mass of our rocket to 0.323g and we obtained a distance travelled of 14.75553m, a few meters away from the designated distance. During the firing test, we collected some data to calculate the coefficient of kinetic friction and the coefficient of aerodynamic friction in order determine the mass we require to hit the target distance.Determining the Coefficient of Kinetic FrictionTo find kinetic friction coefficient, between the rocket and the cable, where f refers to the kinetic friction, m refers to the mass of the rocket and g refers to the gravitational acceleration. By varying the mass used in the experiments as shown in Figure 2 and 3 below, we are able to plot a graph of x1 against x2, where the gradient is .[pic 9][pic 10]
For Figure 2, F = fke1 =  mg[pic 11]k(x1 – L) =  mg[pic 12]m = [pic 13]For Figure 3, T = mgke2 =  mgk(x2 – L) =  mgm = [pic 14][pic 15][pic 16]Figure 2. Simplified free body diagram of a rubber band pulling the rocket along the guided steel cable  Figure 3. Simplified free body diagram of the rocket being suspended by the same rubber bandAs such we can obtain the following equation  [pic 17] [pic 18] [pic 19] [pic 20]Hence, we can then plot a graph of x1 against x2 (refer to Fig. 1 and Fig. 2 in Appendix), obtaining  = 0.2267 and L = 33.9325 cm[pic 21]Determining the Coefficient of Aerodynamic FrictionTo find coefficient of aerodynamic friction acting on the rocket,  , we are dropping the rocket from varying heights and calculating the aerodynamic friction that the object experiences. At terminal velocity, the weight of the CGRPS will be equal to the drag force experienced by the CGRPS. As such we can obtain the following equation, . From the values we obtained from the experiment described above, we can then plot a graph of distance against time. (Refer to Fig.2 in the Appendix for values and Fig.3 in Appendix)[pic 22][pic 23]By referring to Figure 4 below, it is evident that the velocity of the CGRPS reaches a terminal velocity after t = 1s. Base on the equation below, s=13.05t – 8.1036, the terminal velocity is the gradient of the equation which is 13.05 m/s. With the necessary values, the coefficient of aerodynamic friction can be determined.[pic 24][pic 25]where m = total mass of modified rocket (15 marbles and a wooden board)g = gravitational acceleration v = terminal velocity[pic 26]Figure 4. Graph of distance against time at the time after 1 second where the rocket reaches terminal velocityFinal Outcome