In 2012, Tanner Foust and Greg Tracy drove two full size cars around a 6 storey tall looped track (that's roughly 20 metres).
Before watching the video, calculate the speed at which the cars would need to reach at the bottom of the loop to complete the loop without losing contact.
What modelling assumptions have you made?
Watching the video, the cars look surprisingly slow. Let's look at the maths.
At the top of the loop, we want to have a reaction force, so at that point \[R>0\] By Newton's 2nd Law we have \[R+mg=ma\] and since \[a=\frac{v^2}{r}\] this becomes \[m\frac{v^2}{10}-mg=R>0\] After some rearrangement, we arrive at \[v^2>10g\] Using this in the Law of Conservtion of Mechanical Energy, \[\frac{1}{2}mu^2=\frac{1}{2}(10g)+20g\\u^2=50g\\u>0\Rightarrow u=\sqrt{50g}\approx22ms^{-1}\]
That's roughly 50mph.
What about the assumptions?
I assumed a circular loop. Usually, loops are clothoids (forming tear-drop shapes) to reduce strain on mechanical components. Air resistance, driver error and other resistive forces were ignored.
A big assumption was that the car produced no further driving force once it reached the bottom of the loop, and just coasted round. The model could further be refined by looking at the work done by the driving force and using the Work-Energy Principle. With a driving force of 7000N and a mass of 1.4 tonnes, both ballpark figures from a range of online sources, the required speed would reduce to 13.3m/s which is roughly 30mph.
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