So far we have completely neglected air resistance, but even in the low pressure

Question

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.

5. How far can a capsule coast if it starts at full speed before it has slowed to 710 mph?
6. How much energy is lost due to drag when traveling between L.A. and San Francisco?
7. How much power is dissipated by 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.
8. If exactly 40 capsules are operated in the tube at any moment in time, what is the
total power consumption of the hyperloop system.
9. Would it be possible to power the system using only solar energy from panels mounted
above the tube? Explain.
10. How does the energy consumed per rider per mile for the hyperloop compare to a car
with a single driver?
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?
12. How does the calculated minimum turning radius affect the possible routes that a
hyperloop could take if it were built between LA and San Francisco?

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)

Explanation / Answer

5.

a=0.5g = 0.5*-9.8 = -4.9 m/s^2

vi=760mph= 339.8 m/s, vf= 710mph = 317.4 m/s

Use kinematic equan,

vf^2=vi^2 +2ad

Plugging values,

317.4^2=339.8^2-2*4.9*d

d= 1502.2 m

6.

Distance between L.A and San Francisco = r= 559km = 559000 m

E(lost) = F(drag)*r = 320*559000 = 1.79*10^8 J

7.

Power dissipated = F*v = 320*317.4 = 1.02*10^5 W

8.

Total power = Total capsules* Power dissipated for one capsule = 40* Power dissipated for one capsule = 40*1.02*10^5 = 4.08*10^6 W

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