Non Uniform Acceleration - Formula Help

[Hyvin]

Homework Statement

I'm working on a summer project dealing with high altitude balloons and I am trying to come up with a way to calculate the altitude at any given time, but I have not dealt with non uniform acceleration before. The formula I have for acceleration deals with both pressure and temperature, which changes in relation to altitude, but I can't seem to figure out how to calculate an altitude when my answer depends on a previous value of itself (new altitude from acceleration based on old altitude.)

Homework Equations

This is all that I am really looking for right now.

The Attempt at a Solution

I'm assuming that there is some kind of differential equation at work here, but I'm really just not seeing it right now.

 

[rcgldr]

You'll probably need to use some form of numerical method such as Runge Kutta, to solve a second order ODE, to determine velocity and position, given an equaion that calculates acceleration for initial and current velocity and position. This wiki article explains the common RK4 method for a single order ODE. You'll need to expand on this for a second order ODE.

http://en.wikipedia.org/wiki/Runge-Kutta

 

[Chestermiller]

Homework Statement

I'm working on a summer project dealing with high altitude balloons and I am trying to come up with a way to calculate the altitude at any given time, but I have not dealt with non uniform acceleration before. The formula I have for acceleration deals with both pressure and temperature, which changes in relation to altitude, but I can't seem to figure out how to calculate an altitude when my answer depends on a previous value of itself (new altitude from acceleration based on old altitude.)

Homework Equations

This is all that I am really looking for right now.

The Attempt at a Solution

I'm assuming that there is some kind of differential equation at work here, but I'm really just not seeing it right now.

It would be helpful if you provided the equations you are trying to solve. Any chance of that?

 

[Villyer]

So you have a formula for pressure with respect to altitude, a formula for temperature with respect to altitude, and a formula for acceleration with respect to pressure and temperature?

 

[Hyvin]

Thank you all for your quick replies! Here are the equations that I'm up against here as a reminder I'm seeking a Z in terms of variables dependent upon Z. I have initial values for temperature, pressure, and altitude.

It appears Latex doesn't want to play nice for whatever reason with velocity the \frac13 is an exponent to 1/3 and the last fraction with the line under it has everything to the right of it in the denominator.

I'm currently trying to wrap my brain around what rcgldr suggested, but I'm still open to additional suggestions. I would like to say that it seems a bit odd to me that acceleration and velocity do not seem to be related, but it's what I've been given to work with.

Acceleration:
$\frac{d^{2}Z_{b}}{dt^{2}}=\frac{(m_{g}MT_{g}-m_{t}g+F_{z})}{(m_{t}+C_{m}m_{g}MT_{g})}$

Velocity:
$\frac{dz_{b}}{dt}=\sqrt{4(\frac{2}{9\pi})^{\frac{1}{3}}\frac{g}{C_{D}}(\frac{m_{t}}{\rho_{a}})^{\frac{1}{3}}\frac{(1+f)T_{g}-1}-{(1+f)^{\frac{2}{3}}T_{g}^{\frac{2}{3}}}}$

Pressure and Temperature for the moment will be considered to be linear functions of altitude (z)

 

[Chestermiller]

Please define the other variables in your equations. In order to understand this, we first need to know the physical mechanisms that are present, and the physical parameters involved. For example, your second equation looks like it is related to air drag on the balloon. The first equation probably is related to the net buoyancy forces, and the inertial term. Both buoyancy and drag ought to be included in a single equation. Also, it is possible that, for your system, the buoyancy and the drag terms may nearly cancel, so that the inertial term can possibly be neglected. We need to see more detail on how these equations were formulated. You may have a formulation issue that needs to be addressed before we can begin to worry about how to solve the equations.