@acer I like your code in which the plot command itself accomplishes the rotation of colors around the circle as, at the same time, it translates the circle along the x axis.

Please see my reply to tomleslie where I ask whether this can be accomplished by a custom color scheme as portrayed in the colorscheme help page. 

@tomleslie I appreciate your ingenious way of "marching" the colors in arcs around the full circle.

The colorscheme help page contains an example of the colorscheme parameter invoking a separately defined procedure to implement a custom color scheme.

I am intrigued by this but I don't understand what data is communicated from the example's plot3d command to procedure p and what such a procedure must return to the plot3d command.

Can the rotating colored circle be implemented using an analogous method? If possible, then the plot command could itself perform the equivalent rotation and translation.

@Axel Vogt 

@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.

Velodrome.mw

@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.