The Hyperloop is a proposed form of high-speed transportation that accelerates c

Question

The Hyperloop is a proposed form of high-speed transportation that accelerates cylindrical pods that can carry up to 28 passengers in a low pressure tube at speeds of 760 mph. In a whitepaper published by Elon Musk (http://www.spacex.com/sites/spacex/files/ hyperloop_alpha.pdf), a 380 mile route between L.A. and San Francisco is discussed, and the following constraints are introduced: The route should follow existing freeways (i.e. I-5) as closely as possible, the maximum acceleration should be limited to 0.5 g for rider comfort. the maximum speed should be limited to 760 mph to avoid sonic booms. By levitating the pods using either air pressure or magnetism and evacuating most of the air from the tube, the pods can be made virtually frictionless. The pods would be accelerated up to cruising speed using linear induction motors (a line of electromagnets built into the track that act on the pod going over them). For this assignment assume an empty pod has a mass of 3100 kg, and that each rider has an average mass of 65.0 kg. So far we have completely neglected air resistance, but even in the low pressure of the tube, there is expected to be about 320 N of drag. Linear motors would be placed along the track to periodically boost the speed of the capsule to compensate for the loss of speed due to drag. The energy put in to accelerating the capsules to a given speed can be extracted when using the same linear motors as electrical generators when braking. Thus (neglecting any electrical inefficiencies and any energy required to pump air out of the tube) the net energy consumed by the system is entirely due to work done by air resistance.

11. what is the minimum radius that the path of the tube can have as it winds through the terrain if the pods are to maintain cruising speed throughout their journey?

Explanation / Answer

as the maximum acceleration of the pods can be a = 0.5g
and pod speed, v = 760 mph = 339.75 m/s
so let the minimum radius the pod can have be r
then centrifugal acceleration in the pod at the turn of radius r is
ac = v^2/r = 0.5g
r = 339.75^2/0.5g = 23533.1422 m
hence the smallest radius turn that the pod in hyperloop can take is 23533.1422 m = 14.6228 miles

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