Problem of inter continental ballistic missiles

Main Question or Discussion Point

Hello guys I have a problem with projectile motion. Suppose we launch an Inter Continental Ballistic Missile from one point to another on earths surface ( For example from Tokyo to California ) how do we describe the kinematics of the missile.[ The problem is that the "g" vector is rotating and also we cannot choose a linear co-ordinate system. Any ideas ?
 

Answers and Replies

pervect
Staff Emeritus
Science Advisor
Insights Author
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I would suggest using Lagrangian mechanics to solve the problem. Basically you chose a convenient coordinate system (lattitude, longitude, and height? or perhaps Euler angles?), and then you write the Lagrangian in that coordinate system as a function of your chosing variables, and their time derivatives.

For this simple problem, the Lagrangian L of the missile will be the kinetic energy T in an earth-centered inertial frame minus the potential enregy V in an ECI frame.

Then you use Lagrange's equations to get the equations of motion for the missile.

There's an overview at the Wikipedia

http://en.wikipedia.org/wiki/Lagrangian_mechanics

it may not be clear enough if you are not familiar with the subject. You may have to consult a textbook if you want a really detailed explanation. The quick overview is that you have a function L, called the Lagrangian which is written in the form

L(x, x', t), where x is is a coordinate, x' is it's time derivative, and t is time.

Then Lagrange's equations give you the equations of motion directly from the Lagrangian

[tex]
\frac{d}{dt}\left(\frac{\partial L}{\partial x'}\right) =\frac{\partial L}{\partial x}
[/tex]

A simple example - in cartesian coordinates in a potential V with only one coordinate x

L(x,x') = .5*m*x'^2 - V(x)

(note that this is kinetic energy minus potential energy).

Then

d/dt(m*x') = -[itex]\partial V/\partial x[/itex]

For systems with more than one coordinate, there is one Lagrange's equation for each independent coordiante (variable).
 
reilly
Science Advisor
1,075
0
This problem is discussed in many textbooks, and is basic to the study of ballistics. Standard stuff. You want really hard; add a gyroscope to the system and then work out the dynamics of the combined system.

Regards,
Reilly Atkinson
 
Pengwuino
Gold Member
4,854
14
Would this be an application in Differential Geometry?
 

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