Angle for Maximum Range in Projectile Motion with Drag Equation?

[fluxions]

Homework Statement

The problem is Kibble & Berkshire 3.8, http://books.google.com/books?id=0a...kibble berkshire&pg=PA68#v=onepage&q&f=false" is a link. (The problem is on page 68. Equation 3.17, referred to in the problem, is on page 53.)

I figured out the first part (finding the first-order approximation of the range), but I'm having problems figuring out the angle for maximum range.

Edit: I originally said it was problem 3.7, but it's really problem 3.8 I'm having problems with. The link is still valid.

The Attempt at a Solution

The velocity is v^2 = u^2 + w^2 (see page 53). The angle we're looking for is \alpha = arctan(w/u).

It seems like one should be able to write the expression for x in terms of v, and find the extrema of that function. However it is impossible to write x solely in terms of v (and some constants). Also, I don't really understand the hint. Halp!

 

[fzero]

You can certainly express u and w in terms of v and $$\alpha$$. You can then extremize x with respect to $$\alpha$$. As for the hint, if you keep all terms in $$\alpha$$, you arrive at a cubic equation for $$\sin\alpha$$, which is a bit unwieldy. However, any way that I try to use the hint, I'm always off by a numerical factor from their result for $$\cos 2\alpha$$. Maybe you'll have better luck.

 

[fluxions]

Still stuck on this problem.

I can write, using the expression for x given in the link and the relation between u,v,w, and alpha

$$x = \frac{2}{g} v^2 \sin\alpha \cos\alpha - \frac{8 \gamma}{3 g^2} v^3 \cos\alpha \sin^2\alpha$$

Maximizing this function wrt. to v doesn't lead to the desired answer (I'm omit the details).

So I try to use the hint("in the term containing \gamma, you may use the zeroth-order approximation for the angle") Ok, so for \alpha small, the zeroth order approx. of cos(\alpha) is 1 and the zeroth-order approx. of sin^2(\alpha) is ... 0? This doesn't make sense. And clearly it leads nowhere.

Maybe I'm supposed to leave the sin^2 term alone. If I try that, I still don't get the desired answer.

Any insight into how to solve this problem is appreciated.