Designing Transfer Curves for Continuous Train Track Connections

[go_bucks45]

Homework Statement

In designing transfer curves to connect sections of straight railroad tracks, it's important to realize that the acceleration of the train should be continuous so that the reactive force exerted by the train on the track is also continuous. This will be the case if the curvature varies continuously.

A logical candidate for a transfer curve to join existing train tracks given by y = 1, for x<= 0, and y = sqrt(2) - x, for x >= 1/sqrt(2) might be the function f(x) = sqrt(1 - x^2), 0 < x < 1/sqrt(2).

Show that the function:

F(x) =
1 if x<= 0
sqrt(1 - x^2) if 0 < x < 1/sqrt(2)
sqrt(2) - x if x >= 1/sqrt(2)

is continuous and has continuous slope, but does not have continuous curvature. Therefore f is not an appropriate transfer curve.

 

[coomast]

You should show some work on how you tried to solve it, even if it is incomplete.

Hint:
1) How do you mathematically express that two curves are continuous in the point where they touch?
2) The same for showing how they have the same curvature.