Analysis of a water-propelled rocket: A problem in honors physics
18 Feb 2000-American Journal of Physics (American Association of Physics Teachers)-Vol. 68, Iss: 3, pp 223-227
TL;DR: In this article, the authors presented a numerical solution for the height of the rocket, as well as several analytic approximations, and five out of six lab groups predicted the maximum height of a water-propelled, air-pumped, water-powered rocket within experimental error.
Abstract: The air-pumped, water-propelled rocket is a common child’s toy, yet forms a reasonably complicated system when carefully analyzed. A lab based on this system was included as the final laboratory project in the honors version of General Physics I at the USAF Academy. The numerical solution for the height of the rocket is presented, as well as several analytic approximations. Five out of six lab groups predicted the maximum height of the rocket within experimental error.
Citations
More filters
TL;DR: In this paper, a brief review of the physics of toys intends to show that they are not only very useful in lectures and demonstrations in order to motivate students but also very interesting from a scientific point of view.
Abstract: The use of toys in physics teaching is common. This brief review of the physics of toys intends to show that they are not only very useful in lectures and demonstrations in order to motivate students but also very interesting from a scientific point of view. However, since their physics is sometimes too cumbersome, the effect can be the opposite. We call attention to some subtleties of toys used in physics or in general science teaching.
46 citations
Cites background from "Analysis of a water-propelled rocke..."
...Based on that principle, the water rocket [14] is a popular toy with slightly more complicated physics because of its variable mass....
[...]
01 Dec 2010
TL;DR: Details are provided about implementation and evaluation of one PBL project and how difficulties in evaluation of the linked-class PBL experiences are being addressed.
Abstract: Problem-Based Learning (PBL) is a problem-centered teaching method with exciting potential in engineering education for motivating and enhancing student learning. Implementation of PBL in engineering education has the potential to bridge the gap between theory and practice. Two common problems are encountered when attempting to integrate PBL into the undergraduate engineering classroom: 1) the large time requirement to complete a significant, useful problem and 2) the ability to determine its impact on students. Engineering, mathematics, and science professors at West Texas A&M University (WTAMU) have overcome the large time commitment associated with implementation of PBL in a single course by integrating small components of the larger project into each of their classes and then linking these components with a culminating experience for all the classes. Most of the engineering students were concurrently enrolled in the engineering, mathematics, and science classes and were therefore participating in all activities related to the project. This linked-class PBL experience addressed course concepts, reinforced connections among the courses, and provided real-world applications for the students. Students viewed the experience as beneficial, increasing their understanding of content and applications in each discipline. This paper provides details about implementation and evaluation of one PBL project and how difficulties in evaluation of the linked-class PBL experiences are being addressed.
24 citations
TL;DR: The water rocket as mentioned in this paper is a popular toy that is often used in first year physics courses to illustrate Newton's laws of motion and rocket propulsion, and is made of a soda bottle, a bicycle pump, a rubber stopper, and some piping.
Abstract: The water rocket 1 is a popular toy that is often used in first year physics courses to illustrate Newton’s laws of motion and rocket propulsion. In its simplest version, a water rocket is made of a soda bottle, a bicycle pump, a rubber stopper, and some piping see Fig. 1. The bottle is half-filled with water, turned upside-down, and air is pushed inside the bottle via a flexible pipe that runs through the stopper. When the pressure builds up, the stopper eventually pops out of the neck. The water is then ejected and the rocket takes off. Witnesses of the launch of a water rocket cannot but be amazed that such a simple device can reach a height of tens of meters in a fraction of a second. The popularity of water rockets extends beyond physics classrooms, with many existing associations and competitions organized worldwide. 1 The more than 5000 videos posted on YouTube with the words “water rocket” in their title testify to their popularity. Some of these videos involve elaborate technical developments such as multistage water rockets, nozzles that adapt to the pressure, the replacement of water by foam or flour, underwater rocket launches, and even a water-propelled human flight. The public’s passionate explorations with water rockets contrast with the small number of articles devoted to their analysis. I found only two papers 2,3 that treat the simplest possible rocket, similar to
18 citations
TL;DR: In this article, a simple free-fall experiment for undergraduate students to reasonably estimate the drag coefficient of water rockets made from plastic soft drink bottles was performed using relatively small fall distances (only about 14 m).
Abstract: The flight trajectory of a water rocket can be reasonably calculated if the magnitude of the drag coefficient is known. The experimental determination of this coefficient with enough precision is usually quite difficult, but in this paper we propose a simple free-fall experiment for undergraduate students to reasonably estimate the drag coefficient of water rockets made from plastic soft drink bottles. The experiment is performed using relatively small fall distances (only about 14 m) in addition with a simple digital-sound-recording device. The fall time is inferred from the recorded signal with quite good precision, and it is subsequently introduced as an input of a Matlab® program that estimates the magnitude of the drag coefficient. This procedure was tested first with a toy ball, obtaining a result with a deviation from the typical sphere value of only about 3%. For the particular water rocket used in the present investigation, a drag coefficient of 0.345 was estimated.
18 citations
01 Jan 2007
TL;DR: Problem Based Learning (PBL) as discussed by the authors is a teaching method in which students learn through solving a problem, and it has been used to address the problem of lack of motivation in engineering education.
Abstract: Educators and employers share a common goal of “educating/employing people who are highly motivated and who give 100% effort to their work”. The main reason that this goal is not realised is because the people being taught/employed lack motivation. On the education side a lot of what is taught is not applied so the students cannot see the benefit of this learning. On the employment side the workers are not able to apply what they have spent their undergraduate time learning. What is missing? The link between the students’ learning and industry needs. Problem based learning (PBL) seeks to addresses this problem. PBL is a teaching method in which students learn through solving a problem. This paper reports on how PBL addresses engineering education.
8 citations
References
More filters
Book•
01 Jun 1969
TL;DR: In this paper, Monte Carlo techniques are used to fit dependent and independent variables least squares fit to a polynomial least-squares fit to an arbitrary function fitting composite peaks direct application of the maximum likelihood.
Abstract: Uncertainties in measurements probability distributions error analysis estimates of means and errors Monte Carlo techniques dependent and independent variables least-squares fit to a polynomial least-squares fit to an arbitrary function fitting composite peaks direct application of the maximum likelihood. Appendices: numerical methods matrices graphs and tables histograms and graphs computer routines in Pascal.
12,737 citations
TL;DR: Numerical methods matrices graphs and tables histograms and graphs computer routines in Pascal and Monte Carlo techniques dependent and independent variables least-squares fit to a polynomial least-square fit to an arbitrary function fitting composite peaks direct application of the maximum likelihood.
Abstract: Uncertainties in measurements probability distributions error analysis estimates of means and errors Monte Carlo techniques dependent and independent variables least-squares fit to a polynomial least-squares fit to an arbitrary function fitting composite peaks direct application of the maximum likelihood. Appendices: numerical methods matrices graphs and tables histograms and graphs computer routines in Pascal.
10,546 citations
Book•
01 Jan 1966
TL;DR: Inverse Trigonometric and Hyperbolic Functions as mentioned in this paper, the exponential and trigonometric functions of complex numbers are used to define the series of positive terms in the complex number space.
Abstract: Chapter 1: Infinite Series, Power Series.The Geometric Series.Definitions and Notation.Applications of Series.Convergent and Divergent Series.Convergence Tests.Convergence Tests for Series of Positive Terms.Alternating Series.Conditionally Convergent Series.Useful Facts about Series.Power Series Interval of Convergence.Theorems about Power Series.Expanding Functions in Power Series.Expansion Techniques.Accuracy of Series Approximations.Some Uses of Series.Chapter 2: Complex Numbers.Introduction.Real and Imaginary Parts of a Complex Number.The Complex Plane.Terminology and Notation.Complex Algebra.Complex Infinite Series.Complex Power Series Disk of Convergence.Elementary Functions of Complex Numbers.Euler's Formula.Powers and Roots of Complex Numbers.The Exponential and Trigonometric Functions.Hyperbolic Functions.Logarithms.Complex Roots and Powers.Inverse Trigonometric and Hyperbolic Functions.Some Applications.Chapter 3: Linear Algebra.Introduction.Matrices Row Reduction.Determinants Cramer's Rule.Vectors.Lines and Planes.Matrix Operations.Linear Combinations, Functions, Operators.Linear Dependence and Independence.Special Matrices and Formulas.Linear Vector Spaces.Eigenvalues and Eigenvectors.Applications of Diagonalization.A Brief Introduction to Groups.General Vector Spaces.Chapter 4: Partial Differentiation.Introduction and Notation.Power Series in Two Variables.Total Differentials.Approximations using Differentials.Chain Rule.Implicit Differentiation.More Chain Rule.Maximum and Minimum Problems.Constraints Lagrange Multipliers.Endpoint or Boundary Point Problems.Change of Variables.Differentiation of Integrals.Chapter 5: Multiple Integrals.Introduction.Double and Triple Integrals.Applications of Integration.Change of Variables in Integrals Jacobians.Surface Integrals.Chapter 6: Vector Analysis.Introduction.Applications of Vector Multiplication.Triple Products.Differentiation of Vectors.Fields.Directional Derivative Gradient.Some Other Expressions Involving V.Line Integrals.Green's Theorems in the Plane.The Divergence and the Divergence Theorem.The Curl and Stokes' Theorem.Chapter 7: Fourier Series and Transforms.Introduction.Simple Harmonic Motion and Wave Motion Periodic Functions.Applications of Fourier Series.Average Value of a Function.Fourier Coefficients.Complex Form of Fourier Series.Other Intervals.Even and Odd Functions.An Application to Sound.Parseval's Theorem.Fourier Transforms.Chapter 8: Ordinary Differential Equations.Introduction.Separable Equations.Linear First-Order Equations.Other Methods for First-Order Equations.Linear Equations (Zero Right-Hand Side).Linear Equations (Nonzero Right-Hand Side).Other Second-Order Equations.The Laplace Transform.Laplace Transform Solutions.Convolution.The Dirac Delta Function.A Brief Introduction to Green's Functions.Chapter 9: Calculus of Variations.Introduction.The Euler Equation.Using the Euler Equation.The Brachistochrone Problem Cycloids.Several Dependent Variables Lagrange's Equations.Isoperimetric Problems.Variational Notation.Chapter 10: Tensor Analysis.Introduction.Cartesian Tensors.Tensor Notation and Operations.Inertia Tensor.Kronecker Delta and Levi-Civita Symbol.Pseudovectors and Pseudotensors.More about Applications.Curvilinear Coordinates.Vector Operators.Non-Cartesian Tensors.Chapter 11: Special Functions.Introduction.The Factorial Function.Gamma Function Recursion Relation.The Gamma Function of Negative Numbers.Formulas Involving Gamma Functions.Beta Functions.Beta Functions in Terms of Gamma Functions.The Simple Pendulum.The Error Function.Asymptotic Series.Stirling's Formula.Elliptic Integrals and Functions.Chapter 12: Legendre, Bessel, Hermite, and Laguerre functions.Introduction.Legendre's Equation.Leibniz' Rule for Differentiating Products.Rodrigues' Formula.Generating Function for Legendre Polynomials.Complete Sets of Orthogonal Functions.Orthogonality of Legendre Polynomials.Normalization of Legendre Polynomials.Legendre Series.The Associated Legendre Polynomials.Generalized Power Series or the Method of Frobenius.Bessel's Equation.The Second Solutions of Bessel's Equation.Graphs and Zeros of Bessel Functions.Recursion Relations.Differential Equations with Bessel Function Solutions.Other Kinds of Bessel Functions.The Lengthening Pendulum.Orthogonality of Bessel Functions.Approximate Formulas of Bessel Functions.Series Solutions Fuch's Theorem.Hermite and Laguerre Functions Ladder Operators.Chapter 13: Partial Differential Equations.Introduction.Laplace's Equation Steady-State Temperature.The Diffusion of Heat Flow Equation the Schrodinger Equation.The Wave Equation the Vibrating String.Steady-State Temperature in a Cylinder.Vibration of a Circular Membrane.Steady-State Temperature in a Sphere.Poisson's Equation.Integral Transform Solutions of Partial Differential Equations.Chapter 14: Functions of a Complex Variable.Introduction.Analytic Functions.Contour Integrals.Laurent Series.The Residue Theorem.Methods of Finding Residues.Evaluation of Definite Integrals.The Point at Infinity Residues of Infinity.Mapping.Some Applications of Conformal Mapping.Chapter 15: Probability and Statistics.Introduction.Sample Space.Probability Theorems.Methods of Counting.Random Variables.Continuous Distributions.Binomial Distribution.The Normal or Gaussian Distribution.The Poisson Distribution.Statistics and Experimental Measurements.
692 citations
16 citations
TL;DR: The most efficient way to operate a rocket is to increase its exhaust velocity as it accelerates as discussed by the authors, when this increase is done properly, the final kinetic energy of the rocket is maximized.
Abstract: The most efficient way to operate a rocket is to increase its exhaust velocity as it accelerates. When this increase is done properly, the final kinetic energy of the rocket is maximized. It is shown that the resulting ‘‘perfect rocket’’ is far simpler to analyze than the traditional constant‐thrust rocket and provides an excellent application of the material taught in all first semester noncalculus physics courses.
11 citations