# Energy of the rocket

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1. Oct 22, 2014

### sergiokapone

1. The problem statement, all variables and given/known data
How does change the total energy of the rocket during its motion inin a uniform gravitational field?

2. The attempt at a solution
My idea is to write the law of conservation of energy systems - "rocket- gases" $\frac{dE_{total}}{dt}=0$
But get very cumbersome terms , the physical sense , I can not understand.
Besides, it is not clear how to express the change of potential energy of emitted gases .

2. Oct 22, 2014

### BiGyElLoWhAt

I'm sorry, what does this mean?

3. Oct 22, 2014

### sergiokapone

$E=\frac{mv^2}{2}+mgz$

4. Oct 22, 2014

### sergiokapone

I found
$\frac{dE}{dt}=\frac{dm}{dt}(gz+\frac{v^2}{2}-uv)$

5. Oct 22, 2014

### sergiokapone

Ok, my problem is:
Find mass vs time for the rocket, which is a rise in a uniform gravitational field , if the value $-u^2 \dot m$ is constant ?
where $u$ - is the velocity of burned propellant relative to the rocket.
I use
$m \ddot z=-mg-u\dot m$
But I have no idea how to find $m(t)$.

6. Oct 23, 2014

### BiGyElLoWhAt

Why are you treating velocity as constant? I'm pretty sure the only "thing" with constant velocity is the ejected mass from the rocket.
So the force that the ejected matter exerts on the rocket is in the same direction as the force that the earth exerts on the rocket?

7. Oct 23, 2014

### BiGyElLoWhAt

*comment retracted*

8. Oct 24, 2014

### sergiokapone

Of course not, because $-u\dot m$ is positive value, because $\dot m$ - is negative.

Velocity $v(t)$ is unknovn.

9. Oct 25, 2014

### BiGyElLoWhAt

If $u^2\dot{m}$ is constant what do you know about m? Is there any reason why u should vary?
Also, from your energy equation, you have a product on the RHS, you have to treat it as such.

10. Oct 25, 2014

### sergiokapone

It seems to me that the condition of the problem is not correct. For solutions need to know either the $u$or acceleration of the rocket $\ddot z$ .

11. Oct 25, 2014

### haruspex

I believe there is enough information if you assume a constant value for the energy density of the fuel.

12. Oct 28, 2014

### sergiokapone

I solved this promblem and get:

$m=m_0\left[1+\frac{m_0}{2P}\int_{0}^t{\left(\ddot z + g\right) ^2 dt}\right]^{-1}$

I think that if the rocket engine power is constant $-\frac{\dot m u^2}{2}$, the astronaut who has to manage a rocket, need to change the velocity of the propelant $u$ . Thus, the astronaut can control the rocket, ie change the velocity by your own ($\ddot z$). So to solve the problem, we must assume that $u$ (or $\ddot z$) is independent variable. I assumed that $\ddot z$ is such.

13. Oct 28, 2014

### haruspex

I don't see how the astronaut can change the velocity of the exhaust if both the power and the energy density are fixed. Isn't the energy density $\frac 12 u^2$? In other words, it seems to me $\dot m$ is constant.

14. Oct 29, 2014

### sergiokapone

As stated in the problem statement - $\dot m \frac 12 u^2$ is the power of the rocket engine, and it is constant. Thus, astronaut can manage by changing $u$.

15. Oct 29, 2014

### haruspex

Yes, I understand that.
That was not stated in post #5. You added that at post #12 as an opinion. I'm saying that if the energy density is constant then there is no apparent ability to vary u. Let that density be $\rho$:
$\rho (-\dot m) = power = -\frac 12 \dot m u^2$

16. Oct 29, 2014

### sergiokapone

$\rho$ - is not an energy density, because it has dimension $J\cdot kg^{-1} = (m/s)^2$ - specific kinetic energy of the fuel or velocity squared.

17. Oct 29, 2014

### haruspex

I don't understand your point. My $\rho$ is an energy density because I defined it that way. In my equation it has dimension of velocity squared, as you say. Where's the inconsistency?