Continuing our previous post, we have created motion models for a free-falling object with air resistance. For fluid resistance \(F_f=kv^2\), these are:
- \(v(t)=\sqrt{\frac{m g}{k}} \tanh \left(\operatorname{arctanh}\left(v_0 \sqrt{\frac{k}{m g}}\right)-t \sqrt{\frac{k g}{m}}\right)\)
- \(a(t)=-g \operatorname{sech}^2\left(\operatorname{arctanh}\left(v_0 \sqrt{\frac{k}{m g}}\right)-t \sqrt{\frac{k g}{m}}\right)\)
- \(s(t)=s_0-\frac{m \ln \left(\cosh \left(\operatorname{arctanh}\left(v_0 \sqrt{\frac{k}{m g}}\right)-t \sqrt{\frac{k g}{m}}\right)\right)}{k}\)
With these models, we can describe the motion of a free-falling object with drag resistance. Now, let's see how we can exactly apply these models.
Application
Let's say we're designing parachutes. Imagine we're making these life-saving devices. Before we can make one, we need to know what is happening with the falling object.
Say we have an object that fell from the top of a building:
- The falling thing has a mass of 65 kg \(m\)
- We drop it \(v_0=0\) from a height of 800m, initial position \(s_0\), from the ground floor.
- Gravitational acceleration is 9.81 \(\frac{m}{s^2}\)
- Variable \(k\) is equal to 0.035
