How Can Variational Calculus Help Optimize Bullet Design?

[avocadogirl]

AAAAHHH! Calculus of Variations

Homework Statement

See attached

This is a project for an upper level math methods of physics course. My background is insufficient and ultimately, I don't know what is going on, AT ALL. The work I've provided is the product of the collective efforts of my partner, myself and, our supervising professor. I can follow the argument the professor has made but, I could not have made that argument myself.

My partner is exhausted and, I just need some direction with the variational calculus in solving the form we derived for the Euler-Lagrange equation.

Homework Equations

Euler-Lagrange eqution

The Attempt at a Solution

See attached. Thank you all, very, very much.

 

[CompuChip]

I would love to take a look but unfortunately I don't have MS Word on this computer and due to its sensitivity to viruses I prefer not opening your document at all.
Can you make a PDF document (there are many Word-to-PDF printers) or image out of it, if that is possible?

 

[cristo]

Can you make a PDF document (there are many Word-to-PDF printers) or image out of it, if that is possible?

Ta-da... I'm far too kind!

 

[avocadogirl]

I would think that if the length of the bullet was only twice the distance of the max radius, your bullet could only be long and "slender" to an extent but, is there any way in which you could put the radius in terms of the change of the function defining the contour of the bullet? Could you define the radius in terms of the slope of the function? And, if we define the x-axis to be the axis of symmetry, could we define the constants which would determine the position of the function?

From that, given the restrictions of the length being twice the max radius, could we use variational calculus to determine a minimal change in the function? Because, as I see it, if the surface of contact between the bullet and the air particles is almost ( as close as you can get it ) horizontal, that's going to provide the minimal kinetic energy loss, like the way modern rifle bullets are designed.

Thank you both.