# Thrust force of a rocket ejecting mass

1. Sep 29, 2013

### Emspak

1. The problem statement, all variables and given/known data

OK, this seems simple but I want to make sure I am not doing something totally wrong. The problem says: use the conservation of mass of a system of many particles to shoe that the thrust force of a rocket that ejects mass at rate $\frac{dm}{dt}$ is equal to $$F=-v_e \frac{dm}{dt}$$ where $v_e$ is the velocity of the mass ejected.

3. The attempt at a solution

I looked at it this way: momentum is conserved so if we start with mass m of the rocket, mv = k (where k is a constant).

SInce we have a simple differential equation $F=-v_e \frac{dm}{dt}$ it can be integrated as $-m v_e = Ft$. Taking the derivative w/r/t time we get $-m \frac {d v_e}{dt} = F$

That gets us the F=ma part of the equation, showing that that works. But I notice that if momentum is a k (constant) then $-m v_e = k = Ft$.

There's a step I am missing here I think. I feel like I am almost there. Any hep -- and anyone telling me I have approached this in entirely the wrong way -- would be appreciated.

2. Sep 29, 2013

### D H

Staff Emeritus
The momentum of the rocket is *not* a conserved quantity. What you are missing is that to look at things from the perspective of conservation of momentum you need to look at the rocket+exhaust cloud system.

Another issue is how you look at force. There is a direct connection with the conservation laws if you use $\vec F=d(m\vec v)/dt$. However, this means force is a frame-dependent quantity if mass is not constant. Force becomes frame invariant if you use $\vec F = m\vec a$, but now the immediate connection with the conservation laws is lost.

It might help if you look at things from the perspective of an inertial frame that is co-moving with the rocket (i.e., an inertial frame in which the rocket's instantaneous velocity is zero). The F=dp/dt versus F=ma imbroglio vanishes with this choice.

3. Sep 29, 2013

### arildno

A very simple version, alon DH's reasoning, is the following:

Let us move along with the rocket through a tiny time Dt, in which a little mass Dm has been ejected with a velocity, relative to the rocket, v_e.

That little packet of mass has expreniced a momentum change, what we call an impulse, of Dmv_e

Impulse equals Dt*F, so the little packet of mass experienced the force:

F=v_e*Dm/Dt.

Now, by Newton's 3 law, you'll get the result for the thrust force for the rocket!
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4. Sep 29, 2013

### Emspak

Thanks to you both. I felt like this was a much simpler problem than it looked...