Calculating a Parachute Descent

[Lunar_Lander]

Hello,

I'm still working on my balloning project and now I have arrived at the problem of getting a accurate idea of a descent velocity/time. I am sure that the formula to get the velocity is:

[tex]\sqrt{m*g/0.5*\rho*A}[/tex]

Now density changes with altitude, causing a higher velocity at high altitude and a decreasing velocity while closing in on Earth. In the formula, m is the mass of the payload and A is the cross-sectional area of the parachute.

Question: How can I implement [tex]\rho(h)[/tex], that is the density as a function of altitude, into the formula above? Integration over [tex]\rho[/tex]?

My previous approach was to get myself the atmospheric densities for 4000 m-levels out of the 1976 U.S. Standard Atmosphere, to calculate the velocity by putting in each of the densities in turn, and finally to average the velocities. I don't know if this is also OK, but I can imagine that it is less accurate than a analytical solution I am looking for here.

 

[AndyW]

Hi there,

Thank you for your question and for sharing your progress on your project. You are correct in your understanding that the density of the air changes with altitude, which affects the descent velocity of your balloon. To incorporate this into your formula, you will need to use the concept of integration, as you mentioned. Integration allows you to account for the changing density as the balloon descends through different altitudes.

To implement \rho(h) into your formula, you will need to use the integral form of the formula, which is:

v = \int_{h_0}^{h} \sqrt{\frac{m*g}{0.5*\rho(h)*A}} dh

Where h_0 is the initial altitude and h is the final altitude. This integral will give you the average velocity over the entire descent.

To calculate the integral, you will need to have a function for \rho(h), which represents the density as a function of altitude. This function can be derived from the atmospheric density data you have collected or from a standard atmospheric model. Once you have the function, you can use numerical integration methods such as the trapezoidal rule or Simpson's rule to calculate the integral and determine the descent velocity.

I hope this helps and good luck with your project! If you have any further questions, please don't hesitate to ask. Best of luck!