launch_shuttle(航天飞机发射英文版数学建模)

REQUIREMENT 4
Use a software package for numerical solutions of ODE's to solve the equation (or system of equations) from Requirement 3. Demonstrate with your solution that, under the assumed conditions, if an object is thrown away from the earth with a minimum velocity of 36,736 ft/s (or 25,050 mph) it will never return. (The mean radius of the earth is Re = 20.9 x 106 ft.)
© COPYRIGHT 1998 COMAP
MAY BE PHOTOCOPIED FOR CLASSROOM USE
2
Interdisciplinary Lively Applications Project
PA R T I E S C A P E V E L O C I T Y
“WΒιβλιοθήκη Baiduat
will never come back?
For
several decades we have been placing objects in orbit around the earth. The first American to orbit the earth was Astronaut John Glenn. More recently, the shuttle has carried people and cargo into earth orbits. Satellites, shuttle vehicles, and space stations such as the Russian MIR and the International Space station all need to be placed in orbit. This project is designed to explore the physics and mathematics that we need to know in order to get people and objects into orbit around the earth. Mechanics is the branch of physics that deals with how objects move under the action of forces. Isaac Newton formulated some simple laws of mechanics, which describe the mechanical interactions of physical bodies. He also formulated a law, which describes the gravitational influences that bodies have on one another. Taken together, Newton’s Laws of Motion and his Law of Gravitation can begin to help us understand what we need to do in order to put people into orbit.
TO HANDLE SYSTEMS OF NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS
5. USING A GRAPHICS PACKAGE
INTERDISCIPLINARY LIVELY APPLICATIONS PROJECT IS FUNDED BY THE NATIONAL SCIENCE FOUNDATION, DIRECTORATE OF EDUCATION AND HUMAN RESOURCES DIVISION OF UNDERGRADUATE EDUCATION, NSF GRANT #9455980 © COPYRIGHT 1998 THE CONSORTIUM FOR MATHEMATICS AND ITS APPLICATIONS (COMAP)
AUTHORS: JOHN L. SCHARF (CARROLL COLLEGE) FRANK HUGHES (JOHNSON SPACE CENTER, NASA) MANAGING EDITOR: DAVID C. ARNEY EDITORS: JOSEPH MYERS TIM PRITCHETT
CARRY-THROUGH SUBJECTS: DIFFERENTIAL EQUATIONS AND NEWTONIAN MECHANICS MATHEMATICS CLASSIFICATIONS: SYSTEMS OF NON-LINEAR ORDINARY DIFFERENTIAL EQUATIONS, NUMERICAL METHODS, POLAR COORDINATES, AND PARAMETRIC EQUATIONS PHYSICS CLASSIFICATIONS:CLASSICAL NEWTONIAN MECHANICS, KINETICS, AND KINEMATICS DISCIPLINARY CLASSIFICATIONS: MATHEMATICS, PHYSICS, AND AEROSPACE ENGINEERING PREREQUISITE SKILLS: 1 MODELING WITH DIFFERENTIAL EQUATIONS 2.TRANSLATING PHYSICAL STATEMENTS INTO MATHEMATICAL FORM 3. DIFFERENTIAL CALCULUS 4.USING A NUMERICAL DIFFERENTIAL EQUATION SOLVER AND/OR WRITING A NUMERICAL-SOLVER COMPUTER CODE
NSF INITIATIVE: MATHEMATICS SCIENCES AND THEIR APPLICATIONS THROUGHOUT THE CURRICULUM (CCD-MATH)
Going into Orbit: Launching the Shuttle
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INTRODUCTION:
goes up must come down,” is a precept that we accept from an early age, but is it true? Is it possible to throw an object away from the earth fast enough so that it
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M A T E R I A L S 1. Problem Statement (3 Parts) (Student) 2. Sample Solution (Instructor) 3. Notes for Instructor
REQUIREMENT 2
For now you should not worry about the fact that the earth is rotating about its polar axis or that the atmosphere exerts significant drag forces on an object that we attempt to throw away from the earth. Newton’s Second Law states that the rate of change of the velocity (i.e., the acceleration) of an object of constant mass is proportional to the net force acting on it. The constant of proportionality is the reciprocal of the mass of the object. Assuming that we throw an object straight up, apply Newton’s Second Law to write a differential equation (called the equation of motion) that describes the rate of change of the velocity of the object. To support your equation of motion, use a picture showing the forces that act on the object as it flies away from the earth.
Going Into Orbit
Launching the Shuttle
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Interdisciplinary Lively Applications Project
INTERDISCIPLINARY LIVELY APPLICATIONS PROJECT TITLE: GOING INTO ORBIT - LAUNCHING THE SHUTTLE
© COPYRIGHT 1998 COMAP
MAY BE PHOTOCOPIED FOR CLASSROOM USE
Going into Orbit: Launching the Shuttle
3
REQUIREMENT 3
Combine your result from Requirement 2 with the fact that velocity is the time rate of change of position to write a second-order differential equation, or a system of first-order differential equations, that model the motion of the object you are attempting to throw away from the earth. Also specify the initial conditions for your model.
REQUIREMENT 1
According to Newton’s Law of Gravitation, the gravitational force that the earth exerts on an object is proportional to the mass of the object and inversely proportional to the square of its distance from the center of the earth. This is true provided the object is on or above the surface of the earth. Write an expression to describe the gravitational pull of the earth on an object that is at or above the surface of the earth. Use the fact that the weight of an object at the surface of the earth is mg, the mass of the object mass times gravitational acceleration, to determine the constant of proportionality in Newton's Law of Gravitation. Draw a graph that shows how the earth’s gravitational pull on an object varies with its distance from the surface of the earth.