Projectile Motion Problem - Solving for Alpha and Maximum Distance

[PhDorBust]

Problem statement is here: http://www.phys.uri.edu/~gerhard/PHY520/wmex139.pdf

My approach:
$$y = -\frac{gt^2}{2} + v_0 cos( \alpha) t + h$$

Projectile hits ground at:
$$t_{gnd} = \frac{v_0 sin( \alpha ) + \sqrt{ v_0^2 sin^2 ( \alpha) + 2gh}}{g}$$

Now compute derivative of x and solve for alpha:
$$x_{max} = v_0 cos( \alpha) t_{gnd}$$
$$\frac{dx_{max}}{d \alpha} = 0$$

This last step where you solve for alpha is what buggers me. I get a huge quartic expression in alpha that does not simplify, I suspect the problem setter expects you to make some approximation but I can't figure it out.

 

[BruceW]

In the first equation (the one for y), it should be sine.

I see where you're trying to go with your approach. And I think it would work, but maybe you should try a different approach.

Are you familiar with Lagrange multipliers? Because you've got y=0 as the constraint and x as the function to maximise. The functions x and y are each functions of both alpha and t, so the problem looks well suited to the method of Lagrange multipliers.

 

[PhDorBust]

Worked like a charm, good idea.