Learn Physics with a WaterPropelled Rocket


 Peregrine Vernon Peters
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1 Learn Physics with a WaterPropelled Rocket Jong Tze Kian Category: Physics/astronomy education Abstract Number: 430 National Space Agency, Malaysia ABSTRACT: The study of physics can be fun and exciting when flying a model rocket into sky using only compressed air and water ejection. Young people are especially fascinated with the water rocket activity as it scales complex technical systems of a real rocket down to manageable sizes and setups. While the impression of water rocket is often misled as toys, the study of rocket motion is indeed used for decades to inspire students in their learning of physics. This is not one can simply get from existing introductory physics textbooks as the texts describe fundamental physics in general but not the problems related to rocket motion in particular. This paper addresses some of the fundamental physics dynamics problems of an air pumped and waterpropelled rocket that can be easily understood by local students. The study of water rocket motion provides an excellent application for the material taught in the existing school physics syllabus, in which they can be further analyzed and simulated quantitatively as a potential student project. KEY WORDS: physics learning, rocketry physics, rocket dynamics, water rocket INTRODUCTION The rocketry technology is an important space science application. Launching of a large object like space shuttle is a very complex operation, but scientists and engineers have been successfully launching huge space ships into the space orbit due to well understood of the nature laws that explain the motion of an object. To date, each rocket launching mission still holds the basic science laws and it is basically applied in the popular water rocket activity. The basic principle of a water rocket operation is to eject a volume of water from the rocket nozzle by means of compressed air as energy supply to the rocket. Often, the impression of water rocket is misled as toys. In fact, the description of all physics elements involved with water rocket motion gets rather complex and it has been used for decades to inspire students with the study of physical science (G. A. Finney, 2000; Robert A. Nelson, and Mar E. Wilson, 1976; Robert H. Gowdy, 1995). This paper aims at developing a simple model approach that can be used by local students to study the flight performance of a single water rocket launch. The modelling exercise provides a great opportunity for students to bring together their skills in physical reasoning, analysis and numerical methods as well as in the actual water rocket flight test experiments. Before one could analyse and simulate the rocket flight behaviour using numerical computation method, some of the basic physical dynamics problems of a waterpropelled rocket must first be understood and they are described in the immediate section. DYNAMICS PROBLEM This section describes some of the related fundamental dynamics problems such as the
2 thrust, drag and mass of the rocket as a function of time in order to solve the acceleration, velocity and position of a waterpropelled rocket motion. Figure 1 shows the different forces F acting on a water rocket model. Figure 1. The different forces F acting on the water rocket model. F net = F T + F G + F D (1) where F net is the net force acting on a water rocket, F T is the thrust force, F G is the gravitation force and F D is the drag force. The following sections further describe each physical term in the equation (1) that can be solved numerically. Thrust Due to its mass ejection, the thrust, F T, of a water rocket is given by the wellknown Tsiolkovsky rocket equation (G. A. Finney, 2000; Matthias Mulsow, 2011) F T = v e dm dt (2) where v e is the exhaust velocity of the ejected mass and dm is the rate at which the mass is ejected from the rocket. In this case, the mass consists of water mass, m w, where no air leaves the bottle while the water is expelled, and the pressurised air mass, m a, that provides the rocket additional short amount of thrust at the end of water ejection. Hence, the equation (2) can be used to find both the water and air thrust of a water rocket. Since the mass flow rate at the rocket dt
3 nozzle with aperture area, A N, is just the volume flow rate times the density of the water, dm = ρ dv dt w = ρ dt wa N v e (3) with ρ w the water density. The flowing water out of the opening nozzle of a water rocket can be described by the Bernoulli s equation which is considered to be the conservation of energy principle. This can be written as P ρ wv i 2 + ρ w gy = P atm ρ wv e 2 + ρ w gy (4) where P is the pressure inside the rocket, P atm is the atmospheric pressure, v i is the velocity of the water inside the bottle relative to the whole rocket, g is the gravitational acceleration and y is the height of the water. Bernoulli s equation assumes the water flow is incompressible, nonviscous and irrotational. Since the exhaust velocity is very fast as compared to the velocity of the water inside the bottle, the terms describing the flow of water inside the rocket and the pressure difference due to the water height are neglected. Equation (4) can be solved for the exhaust velocity as v e = 2(P P atm) ρ w (5) Obviously, finding the exhaust velocity therefore depends on determining the pressure inside the bottle as a function of time. The solution assumes that the air inside the bottle behaves as an ideal gas and expands isothermally. As known from thermodynamics, when an ideal gas expands from an initial pressure, P o, and volume, V o, to a final state in time of pressure, P, and volume, V, the relation of these variables can be written as P = P o ( V o V )γ (6) with γ the adiabatic coefficient (i.e. 1.4 for diatomic gases). The volume of air, V, inside the bottle at any time can be described as V = V o + V = V o +A N v e t (7) where V is the expanded air volume within an interval time of t. Thus substituting from the equation (6) and (7) to eliminate the V, one get V o P = P o ( V o +A N v e t )γ (8) Once all water is ejected out from the bottle, the air inside the bottle is still pressurized, free to expand and therefore producing additional amount of thrust. Due to its low density, the thrust provided by the air alone is almost negligible. In general, the equation for air exhaust velocity and air mass flow rate are very much the same as previous equations, the only difference is to replace the water density by the air density, ρ a.
4 Gravitation The gravitational force, F G, is one of the important forces affecting the rocket trajectory and it is easily determined by the following equation. F G = m r g (9) where m r is the total rocket mass and g is the gravitational acceleration. The total rocket mass, m r, is the sum of the water mass, m w, the air mass, m a, and the empty rocket mass, m o. Drag An important velocitydependent force, the aerodynamics drag force, F D, can be expressed as F D = 1 C 2 Dρ a Av 2 (10) or F D = 1 2 C Dρ a A v v (11) with C D the drag coefficient, ρ a the air density, A the reference area corresponds to the geometry of interest, and v the velocity of the rocket. The equation (11) assures that the drag force always act against the velocity vector of the rocket by assign a negative sign to the square of v once the velocity, due to descent, gets a negative sign. The drag coefficient is always associated with a particular surface area. For water rocket, the rocket body and the fins are the two main components contributing to the drag. Yet drag is a fairly minor effect, so the precise value of C D is not so critical (Bunnawut Sawamool et al., 2008; G. A. Finney, 2000). Necessary data could be obtained from a number of resources to estimate its drag coefficient. NUMERICAL SOLUTION For simple approach, the Euler method is applied in the numerical solution and its accuracy is highly dependent on the chosen interval time step such as 0.001s. Instead of using highlevel language and interactive environment software such as MATLAB and MATHCAD, the Microsoft Excel worksheet is chosen to perform the numerical computation because it is ready accessible to school students. The solution is performed using the equations described in previous sections. Prior to execute the numerical computation, students must first determine the following set of parameters listed in Table 1 and Table 2. Table 1. List of constants input Description Parameter Value Unit Acceleration of Gravity g ms 2 Standard Atmosphere Pressure Patm Pa Air Density a kgm 3 Water Density w 1000 kgm 3 Adiabatic Constant Drag Coefficient* C D *C D is not an estimated value but rather chosen randomly to show its effect in the simulated result.
5 Table 2. List of necessary physical variables input Description Parameter Value Unit Initial Internal Pressure P Pa Nozzle Diameter dn m Rocket Body Diameter dr 0.09 m Rocket Volume Vr m 3 Empty Rocket Mass mo kg Water Mass mw 0.7 kg Some of the other necessary parameters input can be derived from following equations that are useful in the numerical computation. m wn = m wn 1 ( dm ) t dt n P m a = ρ a V o o P atm m w = 0 m an = m an 1 ρ a A N v e t before Water Out after Water Out V n = V n 1 + ( m w n 1 m wn ρ w ) (12) a n = F net n m rn v n = v n 1 + a n t y n = y n 1 + v n t Using these parameters, one will be able to perform numerical computation and analyze the characteristics of a water rocket flight performance via computer simulated graphical presentation. The result of the simulation can be further compared with the actual performance of a water rocket launch in the school field. RESULT AND DISCUSSION According to the simulated result, the thrust phase occurs extremely fast as it lasts only less than 100 milliseconds. Figure 2 shows the simulated force curve during the thrust phase. The corrected net force (solid line) is the total force acting on the rocket that has taken the effects of gravity and air drag. Undoubtedly, the gravity and air drag contribute fairly small effect to the total force during the thrust phase. Noticeably, the thrust phase can be separated into two parts: the water ejection phase (water thrust phase), and the exhaust of the remaining pressurized excess air (air thrust phase). During the water thrust phase, the rocket accelerates tremendously to its maximum up to 50 times gravity acceleration at the event of Water Out (see Figure 3). Then the remaining pressurized excess air is exhausted in the air thrust phase. During this phase, the rocket speeds up to its maximum velocity while the corrected net force, the acceleration and the exhaust velocity curve down to zero at the event of Excess Air Out.
6 Figure 2. The force curve during the water and air thrust phases. i) net force corrected with gravity and air drag (solid line); ii) net force corrected only with gravity (wide dashed line); iii) thrust only (short dashed line). Figure 3. The acceleration (solid line) and exhaust velocity (dashed line) during the water and air thrust phases. Subsequently the coast phase of the water rocket begins from the event of Excess Air Out and the flight is slowing down to zero at the maximum altitude (see Figure 4). The descent phase lasts longer in a water rocket flight as it is characterized by a constant descent velocity due to the interaction between the gravity and air drag effects. Since only gravity and air drag forces are acting in both the coast and descent phases, the rocket flight is similar to a basic parabolic trajectory.
7 Figure 4. The maximum altitude (solid line) and rocket velocity (dashed line) during the coast and descent phases. CONCLUSION This paper addresses some of the physics dynamics problems related to the water rocket motion. The described mathematical models are possible used to analyse and simulate the water rocket flight behavior using only simple numerical computation approach. It provides an excellent application for the study of physics that is not one can simply get from existing introductory physics textbooks. Therefore, proper outlined procedures for model development, simulation and investigation of the water rocket flight behavior could lead this to a potential student project that bring together students skills in physical reasoning, analysis and numerical methods. The simulated results can also be used to compare with the actual rocket flight tests performance. Although the analysis and prediction approach maybe beyond the ability of many local students, this would be a rewarding project for those who willing to take the challenge. REFERENCES Bunnawut Sawamool, Pongsathon Somsrida, and Aleck Lee, Optimizing the Performance of Water Rocket with Drag, The 4 th Aerospace Conference of Thailand, King Mongkut s University of Technology North Bangkok (March 24, 2008) G. A. Finney, Analysis of a waterpropelled rocket: A problem in honors physics, American Journal of Physics, vol. 68 (3), p (March 2000) Matthias Mulsow, Modelling and optimization of multistage water rockets, Bachelor Thesis, MathematishNaturwissenschaftliche Fakultat, ErnstMoritzArndtUniversitat Greifswald (July 12, 2011) Robert A. Nelson, and Mar E. Wilson, Mathematical Analysis of a Model Rocket Trajectory. Part I: The powered phase, The Physics Teacher, p (March 1976) Robert A. Nelson, Paul W. Bradshaw, Matthew C. Leinung, and Hugh E. Mullen, Mathematical Analysis of a Model Rocket Trajectory. Part II: The coast phase, The Physics Teacher, p (May 1976) Robert H. Gowdy, The physics of perfect rockets, American Journal of Physics, vol. 63 (3), p (March 1995)
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