The physics of a missile with uniform rate of fuel loss

[Pefgjk]

Hello everyone!

I'm at a loss trying to figure out the specific force that pushes a missile at rest off the ground, to the high sky.

Considering a missile with constant mass, and an independent initial total fuel mass. The rate of fuel mass being expelled out of the missile is constant, in (kg/s); the fuel gas exits the missile at a constant speed wrt to the missile, in (m/s).

For convenience, I'm considering a 1-D case, along the height axis only.

My question is, the calculation of the force that points upward, pushing the missile through the air, with regard to the extent of Newtonian mechanics only.

I have tried applying Lagrangian mechanics to find the force, using the Lagrange-Euler equation for the case with external force, but the result did not seem consistent.

Please, help me if you have the time!

Thanks in advance!

 

[Nugatory]

Google for "Tsiolkovsky rocket equation" to see how the math works out. (If you've already tried Lagrangian methods, you have more than enough math to handle Tsiolkovsky's stuff).

 

[Yan_Dalton]

I don't have much time right now since its 1am here, but summing it up:
The Tsiolkovsky rocket equation (also known as ideal rocket equation) describes the motion of vehicles which follow the basic principles of rockets: basically it describes devices that can apply acceleration to themselves by using thrust by expeling part of its mass with high velocity (in other words, they move due to conservation of momentum).

The equation relates the delta-v with with the effective exhaust velocity and the initial/final mass of the rocket, as shown below.

As Nugatory said, if you already used the Lagrangian methods you should be able to use that.