Projectile motion with air resistance proportional to velocity

[dykkms]

Homework Statement

The projectile with initial velocity v_0 is moving in environment with air resistance, which is linear to projectile's velocity. The task is to prove, that horizontal distance is maximal, if the elevation angle (\alpha) and the angle of trajectory tangent in place of impact are complementary.

Homework Equations

F_x = - k.v_x
F_y = - m.g - k.v_y
v_x = v.\cos(\varphi)
v_y = v.\sin(\varphi)
v_{0x} = v_0.\cos(\alpha)
v_{0y} = v_0.\sin(\alpha)

The Attempt at a Solution

equations in x direction:
a_x = \frac{F_x}{m} = - \frac{k}{m} v_x
a_x = \frac{\mathrm{d}v_x}{\mathrm{d}t} = - \frac{k}{m} v_x
\frac{\mathrm{d}v_x}{v_x} = - \frac{k}{m}\mathrm{d}t \Rightarrow v_x = v_{0x}.e^{-\frac{k}{m}t}
after integration:
x = -\frac{m}{k} v_{0x} . e^{-\frac{k}{m}t} + \frac{m}{k} v_{0x}

equations in y direction:
a_y = \frac{F_y}{m} = - g - \frac{k}{m} v_y
a_y = \frac{\mathrm{d}v_y}{\mathrm{d}t} = - g - \frac{k}{m} v_y
\frac{\mathrm{d}v_y}{- g - \frac{k}{m} v_y} = \mathrm{d}t
after integrations and mathematical expressing:
v_y = \frac{m}{k}\left(g + \frac{k}{m}v_{0y}\right)e^{-\frac{k}{m}t} - g\frac{m}{k}
y = - \frac{m^2}{k^2}\left(g+ \frac{k}{m}v_{0y}\right).\left(e^{-\frac{k}{m}t} - 1\right) - g\frac{m}{k}t

And here is my problem:
I would like to express from y equation the time of the impact t_i. I know about this moment, that y = 0. If I am able to write explicit function for t_i(\alpha), then I would pass it to x function. Afterwards I would differentiate this x function according to \alpha. Then I would find the ideal \alpha, which I would use to find t_i which I would finally use to find impact angle.
Unfortunatelly, I can't see the way to find explicit expression of t_i from y = 0 function.

 

[gabbagabbahey]

Basically, you want to maximize x(t_i)-x(0) subject to the constraint y(t_i)=0...that seems like a prime candidate for the mathod of Lagrange multipliers to me

 

[dykkms]

Basically, you want to maximize x(t_i)-x(0) subject to the constraint y(t_i)=0...that seems like a prime candidate for the mathod of Lagrange multipliers to me

Thank you for your answer!
Basically, you say (and I think you are right...), there are two other possibilities:

• from function y = 0 not to express t_i(\alpha), but \alpha(t_i) and to use this in function x(t_i), which I should maximize
• to use method of Lagrange multipliers

First mentioned method brings me to situation, that I am not able to express t_i, which would maximize x(t_i) function, anyway.
Second, I was thinking about method of Lagrange multipliers before, but I tried to avoid it, because I am trying to help with this homework to my friend, who is first-year student at university (industrial management) and he doesn't know the method yet... But I think, I will have to.

 

[gabbagabbahey]

Hmmm... if this is supposed to be a 1st year level problem, there may be a simpler method...perhaps energy conservation would make things easier.