How Accurate Is My Model for Rocket Displacement with Mass Loss?

[rwalrus]

I'm trying to model (purely for my own edification) the behavior of a rocket (specifically w.r.t. significant mass loss over time). I *think* I have everything correct, but I'm not totally confident with how I'm computing the vertical displacement (assuming no air resistance)

I have the following:
$$m(t) = m_{rocket+fuel} - \dot m t$$
$$v_{rocket}(t) = v_{eject} \cdot ln \left ( m_{rocket+fuel} \over m(t) \right) - g t$$
$${\dot m} = {F_{thrust} \over v_{eject}}$$
$$E_k(t) ={{ m(t) \cdot v(t)^{2} } \over 2}$$

Hopefully so far I'm ok...I'm assuming constant thrust force and eject velocity, which I am given to understand is ok. I'm also neglecting air resistance.

My reasoning is that since I'm starting from 0 velocity, 0 kinetic energy (I know, rotation of the earth, etc. la la la I can't hear you ), the total kinetic energy is the work done, so I have distance:
$$d(t) = {E_k(t) \over F_{thrust}}$$

Is there a better way? I'm working (slowly) on integrating v(t), but it's really quite hairy. And my calc (like my physics) is, well, very poor. (i.e. self taught). I have a spreadsheet where I've got some numbers generated by the above, so if it would be helpful to post those, I can do so.

 

[Aaron Bex]

Your approach to computing vertical displacement is correct. The equation you have will give you the distance traveled by the rocket as it accelerates due to thrust and loses mass. However, you should also consider the fact that the vertical velocity of the rocket will decrease as the rocket ejects fuel and its mass decreases (due to the fact that less force is being applied). To account for this change in velocity, you can integrate the equation for velocity over time. This will give you the total vertical displacement of the rocket.

The equation for velocity can be written as:
v(t) = v_eject ln(m_0/m(t)) - gt

The total vertical displacement can then be found by integrating this equation over time, giving:
d(t) = v_eject (m_0/m(t) - 1) - gt^2/2

This equation will give you the total vertical displacement of the rocket, taking into account both the acceleration due to thrust and the decrease in velocity due to mass loss.

 

[charng]

I would suggest that you first review the basic principles of rocket propulsion and the equations that govern it. This will help you understand the behavior of a rocket and how to accurately model its displacement.

One important factor to consider is the change in mass over time due to fuel consumption. This can significantly affect the thrust and velocity of the rocket, and therefore must be included in your calculations. You may also need to consider the effect of air resistance, which can have a significant impact on the rocket's displacement.

Additionally, it may be helpful to use a numerical integration method, such as the Euler method, to accurately calculate the rocket's displacement over time. This method takes into account the changing variables and can provide a more accurate representation of the rocket's behavior.

Overall, I would recommend consulting with a physics expert or conducting further research to improve your understanding and modeling of rocket displacement. This will ensure that your calculations are accurate and reliable.