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These are replies submitted by Earl

@vv This integral is from Problem 14 in Paul J. Nahin's book How to Fall Slower than Gravity.

@Ramakrishnan Much appreciated.

@acer Both for your advice re Maple Player and better coding

@bmartin Thank you for the hint.

The Lagrangian approach outlined in the Wiki article works well with a pivot moving along a variety of 3D spacecurves.

@Kitonum Please explain why you can ignore the denominators in the solutions to Eq1 and Eq2.

Both your answer and Acer's are enlightening, but sadly i can't designate both as the best answer.

@acer Please explain why freezing the derivatives is necessary for eliminate to work.

Both your answer and Kitonum's are enlightening, but sadly i can't designate both as the best answer.

Your intimate knowledge and use of Maple capabilities is, as ever, very impressive!

@Rouben Rostamian  Thanks for this clarification.

@Rouben Rostamian  Thank you Rouben.

Now I just have to figure out the ODE's which show  a car moving at constant velocity horizontally at the inner edge of the straight section and then sliding up the bank of the curved section until friction, gravity and the car's reduced horizontal (now circular) speed combine to produce just enough centrifugal force to maintain a constant height path around the curved section.

@Christopher2222 Attached is a crude approximation of a banked velodrome track.

I followed the leads you provided in your answer above. I even emailed the Mattamy velodrome in Milton, Ontario but none of these led me to the math which describes a track profile. (Mattamy never answered).

In the unlikely event that I come across such math it will likely be over my capabilities, but it's fun to try.


@vv Thank you vv. I much appreciate the many times you have helped me. 

@vv Your animation looks like the effect I am seeking. Please provide the worksheet that contains it.

@Rouben Rostamian  I will try to apply the technique you have shown me to my Hill profile extended to a surface of revolution i.e. to a banked circular race track with a block sliding around it.

If that succeeds I'll try applying it to a velodrome shape i.e. an oval track with banked semi-circular ends whose slopes taper into level straight aways. That is provided I can figure out a mathematical description of this more complicated shape.

@Christopher2222 Thanks for these references. I have some experience with a clothoid as a vertical element of a roller coaster so I will explore that as a transition curve.

Below is a worksheet containing a crude attempt to display a banked track end joined to a straightaway whose banking increases to the end track's banking as it extends towards the latter. However the straightaway does not approach the banked end on a straight line ( the y axis) as it should.

Is there a way to define the TransitionTrack to correct this?



@Carl Love I read Wikipedia's description of logarithmic differentiation and learned a lot. Much appreciated.

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