Cushioning Effect of Straw at Terminal Velocity

[TheOneAndOnlyX]

Hi there!

For a school science project, we had to research a physics concept relating to a real life scenario. I ended up researching individuals who have fallen out of airplanes and survived the fall, despite travelling at terminal velocity. As such, I decided to research how much hay / straw would be needed to cushion such a fall from the height of a commercial plane.

Firstly, I tried using conservation of energy, with gravitational potential energy being converted into elastic potential energy (assuming that the hay acted as a spring). However, I was unable to find any details regarding the spring constant of hay / straw, and as I'm in lockdown, am unable to calculate it using experimental data that I could gather. Is there any way that I'd be able to find this?

Another approach that I took was regarding the bulk modulus of straw. I used the attached equation to try and calculate how much hay would be needed based on the individual decelerating at differing speeds (as pressure is directly related to Force, and as F = ma, acceleration) ranging from 5g - 20g. Within the equation, B is the Bulk Modulus, Delta P is change in pressure, VI is initial Volume (before compression occurs due to impact of the individual) and VF is final volume (after compression occurs). However, with the bulk modulus that I found for straw (appx 5000Pa), I ended up getting a negative volume for the hay when I rearranged the equation. I don't understand why this occurs.

Finally, I found a university research project which covered pretty much my exact topic (and can be found at https://journals.le.ac.uk/ojs1/index.ph ... /2211/2115). However, I don't understand the integral that they perform when going from Equation One to Equation Three.

Any help would be greatly appreciated regarding any of these methods! If there's anything else that needs to be explained, then I'm happy to send through more of my working!

[charlesg]

Hi TheOneAndOnlyX,

Sounds like an interesting question and phenomenon. I'll try to answer the 3 parts of your question, although any other experts should feel free to step in:

1) Spring constant of hay: The spring constant will vary a lot depending on straw orientation and straw type, so you may want to make some simplifying assumptions. http://iranarze.ir/wp-content/uploads/2017/02/E3643.pdf includes experimental stress/strain data that may be useful.

Another method would be to use a physics engine (like Unity) that already includes accurate individual-straw models. Then you might be able to create simulations of objects falling into bundles of these straw-models. This would require a strong understanding of Unity programming.

2) Bulk modulus of straw: Bulk modulus of straw: (correction from earlier response) I believe there should be a negative in front of the equation (if you consider deltaP = P_i - P_f. They may be assuming the reverse order in that equation).

3) https://journals.le.ac.uk/ojs1/index.ph ... /2211/2115 . Attached are the steps to derive Equation 1 from Equation 3.

Hope that helps. Feel free to let us know if you have additional questions.

Best,
Charles