Answer: Energy of Rocket in Uniform Gravity Field

[sergiokapone]

Homework Statement

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 .

 

[BiGyElLoWhAt]

How does change the total energy of the rocket during its motion inin a uniform gravitational field?

I'm sorry, what does this mean?

 

[sergiokapone]

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

 

[sergiokapone]

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

 

[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 $m(t)$.

 

[BiGyElLoWhAt]

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

Why are you treating velocity as constant? I'm pretty sure the only "thing" with constant velocity is the ejected mass from the rocket.

Ok, my problem is:
$m \ddot z=-mg-u\dot m$

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?

 

[BiGyElLoWhAt]

*comment retracted*

 

[sergiokapone]

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?

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

Why are you treating velocity as constant?

Velocity $v(t)$ is unknovn.

 

[BiGyElLoWhAt]

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 $m(t)$.

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.

 

[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 $\ddot z$ .

 

[haruspex]

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 $\ddot z$ .

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

 

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

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.

 

[haruspex]

I solved this problem 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 $u$ .

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.

 

[sergiokapone]

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.

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 $u$.

 

[haruspex]

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

Yes, I understand that.

Thus, astronaut can manage by changing $u$.

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 $\rho$:
$\rho (-\dot m) = power = -\frac 12 \dot m u^2$

 

[sergiokapone]

I'm saying that if the energy density is constant then there is no apparent ability to vary u.

$\rho$ - 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.

 

[haruspex]

$\rho$ - 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.

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?