Rotating Cockpits in Vehicles

I stumbled across this video by Aren Khachatryan showing an innovative rotating cockpit idea to reduce the impact of head on collisions through circular redirection.

Which led to the following question for students. It lends itself nicely to modelling restrictions (the human being a particle for example) and Aren goes into much more detail here, showing the maths behind it, providing some super-curricular material for students to explore.

Solution

The initial speed is \(72 kmh^{-1} = 20 ms^{-1}\). Take directly below the centre to be \(\theta=0\).

By the convservation of mechanical energy, \(\frac{1}{2}(100)(20^2) =\frac{1}{2}(100)(v^2) + 100g(0.5 - 0.5cos\theta)\)

Resulting in \(v^2=400+gcos\theta -g\)

Then \(a=\frac{v^2}{r}=\frac{400+gcos\theta -g}{0.5}=800-2g+2gcos\theta\)

This has a maximum value of \(800\) when \(\theta=0\) which is roughly \(80 g_0\).

Note and further reading

I saw Aren Khachatryan's idea and thought it would make an interesting problem to look at. The figures in my solution are of a crude model to make it more accessible to students learning about circular motion. They are not a review of Aren's work. His work is much more in depth, taking into account decelerating centre, multiple bodies and modelling considerations. It makes for an interesting read.

While doing some background digging for this, I also came across a car crash calculator that has some explanations with accessible maths for A-Level students.