Hyperloop: Height difference

The Hyperloop is a hypothetical new fast transport system between cities, which works by launching pods that carry people through a very low air pressure tunnel. The Hyperloop will need to change elevation as the pod travels. Imagine you are designing a tunnel for the Hyperloop that has to drop in elevation by some height \(h\) over a horizontal distance of a kilometer. In other words the tunnel will be horizontal at \(x=0~\mbox{m}\) and at height \(h\), then begin to drop in elevation until \(x=1000~\mbox{m}\), at which point the tunnel is horizontal again but at height \(0\).

You decide to make the shape of the tunnel a cubic spline. One thing you need to ensure for safety is that the pod always remains on the bottom of the tunnel supported by the air cushion - smashing into the ceiling of the tunnel is not a good thing. What is the maximum value of \(h\) in meters that can be safely accommodated by such a tunnel?

Details and assumptions

• The pod travels at a constant \(300~\mbox{m/s}\) through the tunnel.
• Model the pod as a point mass.
• In the Hyperloop design there is very little space between the pod and the tunnel.
• The acceleration of gravity is \(-9.8~\mbox{m/s}^2\).