# 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 $$-9.8~\mbox{m/s}^2$$.
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