Rocket Speed: Calculating from Ciolky's Number C

[Garunekk]

Homework Statement

Hi, I would like to apologie for my bad english, but I will try to write my problem as good as I can.
I have a homework...From one stage rocket with initinial speed [m[/0] Ciolky´s number C are escaping gasses with the speed u.
Let´s assume that weight of the rocket is changing by the formula m=m0*e^(-kt).
Calculate the speed of the rocket in general distance r from the middle of the Earth. Gravity acceleration is changing by the formula a=gR^2/r^2

Homework Equations

To be honest, i really have no idea where to start, and I would really apreciate your help.

The Attempt at a Solution

 

[haruspex]

Ciolky´s number C

I assume this is a reference to Ciolkowski, but I am unable to find a definition of a number named after him. Is it perhaps the log of the mass ratio, initial to final?

 

[Garunekk]

Yes, it is. The result should be

But I have no idea how to get that. I tried it through the Ciolkowski formula, but ..I just do not know..

 

[haruspex]

Yes, it is. The result should be

But I have no idea how to get that. I tried it through the Ciolkowski formula, but ..I just do not know..

Ok. But there seems to be something missing here:

with initinial speed [m[/0]

Should it say "initial speed u₀, initial mass m₀"?
When at distance r, what are the forces acting on it? What is the net force? Write some differential equations for distance and mass.

 

[Garunekk]

r, what are the forces acting on it? What is the

Yes, my bad, initial mass, sorry.

The forces acting on it? Gravity force, I guess. ..I am not pretty sure what do you mean by the "net force". And some differential equations..

I got through d/dt=-km0e^(-kt) to m(dv/dt)=-mg+kmu, but, i am lagged now.

 

[haruspex]

what do you mean by the "net force".

There is a gravitational force and a thrust from the engine. The vector sum of these is the net force.

m(dv/dt)=-mg+kmu

how do you get kmu?

 

[Garunekk]

There is a gravitational force and a thrust from the engine. The vector sum of these is the net force.

how do you get kmu?

Maybe I am totally already useless today, but I got similar example in my study textbook (its from my university, and in Czech language, but i try to post it)

so I tried to solve it in this way, but I am stuck. Hope you will understant that, its just series of the formulas, and the "v" should be the result. But when I try to make my example by that way, i get nowhere. In my example is just lambda=k, and vr=u.

 

[haruspex]

Ok, I just wanted to see how you got m(dv/dt)=-mg+kmu.
Now, that g is the gravitational acceleration at radius r, so should really be written g_r to distinguish it from g at Earth's surface. Rewrite it in terms of the surface g, Earth's radius R, an the rocket's radius r.

 

[Garunekk]

Ok, I just wanted to see how you got m(dv/dt)=-mg+kmu.
Now, that g is the gravitational acceleration at radius r, so should really be written g_r to distinguish it from g at Earth's surface. Rewrite it in terms of the surface g, Earth's radius R, an the rocket's radius r.

I really appreciate that you help me, but I don't understand you now. I can rewrite ag as ag=H(Mz)/R^2 ...where H is gravity constant and Mz is weight of the earth.

I understand why result looks like it looks. Except the "2".I have no idea where I got it from. But I do not know how to get that result, or from what.

 

[haruspex]

I can rewrite ag as ag=H(Mz)/R^2 ...where H is gravity constant and Mz is weight of the earth.

No.
You wrote that the gravitational force on the rocket is mg. What did you mean by g there? If you meant the gravitational acceleration at Earth's surface then that is wrong. The force will diminish as r increases. What will it be in general?

 

[Garunekk]

No.
You wrote that the gravitational force on the rocket is mg. What did you mean by g there? If you meant the gravitational acceleration at Earth's surface then that is wrong. The force will diminish as r increases. What will it be in general?

Here is 2am, and I am mad at myself, sorry. I know it will diminish as r increase. I can write is an (r-R), assuming R is the middle of Earth. Do you mean this?

 

[haruspex]

Do you mean this?

No. For a uniform sphere mass M radius R, at a distance r>R from its centre the gravitational field is as though it were a point mass. What is the field at distance r, according to Newton?