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

Related Classical Physics News on Phys.org
pervect
Staff Emeritus
Science Advisor
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

$$\frac{d}{dt}\left(\frac{\partial L}{\partial x'}\right) =\frac{\partial L}{\partial x}$$

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') = -$\partial V/\partial x$

For systems with more than one coordinate, there is one Lagrange's equation for each independent coordiante (variable).

reilly
Science Advisor
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
Would this be an application in Differential Geometry?