A paper on the ‘Hyperloop Alpha’ concept for a high speedtransportation system was published in 2013. Since then, there hasbeen a lot of hype, time, and money directed toward the concept.Your task this week is to evaluate the hyperloop concept purelyfrom a 2D particle kinetics point of view.

For this assignment, the important specifications from the paperare as follows:

Urban cruise speed: 300 mph (480 kph)

Inter-city cruise speed: 760 mph (1,220 kph)

Axial Acceleration (along direction of travel): 1 g

Lateral Acceleration (normal to direction of travel): 0.5 g

Capsule weight (incl. passengers): 57,000 lb (26,000 kg)

Pylon support spacing: 100’ (30 m)

3. At a minimum, any motion path must be continuous in position,velocity, and acceleration to avoid impacts. Suppose the pathdeviates from the theoretical line by a distance h, due to arelative displacement of one pylon with respect to its neighbors.If we assume a polynomial path along these two pylon spacings, fora total distance of 2L, the lowest-acceleration shape that meetsthese end-conditions is given by the function:

y(x) = 64h((x/2L)^3-3(x/2L)^4+3(x/2L)^5-(x/2L)^6)

c. Assuming that the velocity in the nominal motion direction (˙x) is constant, so that v = ˙xˆI + ˙yJˆ, and assuming that y˙ ≪ v,so that y/v ˙ ≈ 0, determine an expression for the tangentialvelocity, the tangential acceleration, and the normal accelerationof the capsule as a function of the amount of deviation (h), thepylon spacing (L), and the nominal velocity ˙x.

d. At what position x does the peak value of the normalacceleration, an occur?

e. At the location of peak normal acceleration, you determinedin part ‘d’, determine an expression for the radius of curvature ρof the motion of the capsule as a function of the amount ofdeviation (h) of the pylon.

g. Could you allow the same h in a curved section of track? Whyor why not?

h. Does this level of precision in straightness sound plausible?Based on the values you calculated, would you expect there to beany significant impact on the cost?