Projectile Motion: Angle & Range with Air Resistance and Wind

[ficku1]

Homework Statement

A projectile is launched at an angle to the horizontal and rises upwards to a peak while moving horizontally. What is the angle of maximum range and how it is dependent on initial velocity if we include air resistance and if the wind is blowing in the horizontal direction of flight?

Homework Equations

The Attempt at a Solution

I used Newton's second law and i took into account quadratic air resistance and that air resistance is in the opposite direction of flight and i got these two equations but i don't know how to solve them. Is this even correct? Do you have any advice on how to solve these two?

a_x= -K(v_x^2+(v_y -w)² )*cosφ
a_y= -K(v_x^2+(v_y -w)^² )*sinφ⁡ - g

Thank you for your help.

 

[Ray Vickson]

Homework Statement

A projectile is launched at an angle to the horizontal and rises upwards to a peak while moving horizontally. What is the angle of maximum range and how it is dependent on initial velocity if we include air resistance and if the wind is blowing in the horizontal direction of flight?

Homework Equations

The Attempt at a Solution

I used Newton's second law and i took into account quadratic air resistance and that air resistance is in the opposite direction of flight and i got these two equations but i don't know how to solve them. Is this even correct? Do you have any advice on how to solve these two?

a_x= -K(v_x^2+(v_y -w)² )*cosφ
a_y= -K(v_x^2+(v_y -w)^² )*sinφ⁡ - g

Thank you for your help.

What is $\varphi$ here? If it is the initial angle, it should not appear in the expressions for the acceleration components; it should just act as an "initial condition" when solving the resulting differential equations. If, on the other hand, $\varphi$ represents the slope of the tangent to the flight curve at a point, then you need to express $\cos \varphi$ and $\sin \varphi$ in terms of $v_x, v_y$.

BTW: the differential equations are likely nasty enough to need numerical solution; that is, there is probably no closed-form formula for the solution of your problem. You might need to resort to a numerical DE-solving package, solving the problem for various angles and plotting the results. Maybe some "reasonable" symbolic approximations can be made that would be adequate over limited ranges of initial speed.

 

[ficku1]

Thank you very much for your answer.

Yes, φ is the initial angle. But isn't the acceleration a_x=-Kv²*cosφ and this equals
a_x= -K(v_x²+(v_y -w)² )*cosφ and this equals a_x= -K(v_x²+(v_y -w)² )^(1/2)*v_x?

I am aware that there isn't any formula for the solution, but I don't know how to continue. If I understand correctly, I have to express the range so that then I can calculate dx/dφ=0 (x is range, we want x( y=0)) and get maximum angle like it is in the case with no air resistance. Is there even possible to get x and y out of the equations above? Which numerical method would you sugest?

Thank you for your help.

 

[Ray Vickson]

Thank you very much for your answer.

Yes, φ is the initial angle. Oh, it is actually hidden in v_x+v_y, sorry.

I am aware that there isn't any formula for the solution, but I don't know how to continue. If I understand correctly, I have to express the range so that then I can calculate dx/dφ=0 (x is range, we want x( y=0)) and get maximum angle like it is in the case with no air resistance. Is there even possible to get x and y out of the equations above? Which numerical method would you sugest?

Thank you for your help.

If you have a numerical value for $K$ and a given initial speed $v_0$, wind speed $w$ and initial angle $\varphi_0$, then you obtain a pair of coupled nonlinear differential equations to determine $v_x(t)$ and $v_y(t)$ for $t > 0$ with known initial conditions. You can then perform numerical integrations to get $x(t)$ and $y(t)$. Alternatively, you can write nonlinear coupled second-order DEs for $d^2x/dt^2$ and $d^2 y/dt^2$ in terms of $dx/dt$ and $dy/dt$, then solve those numerically for given $x(0), y(0), dx/dt|_{t=0}$ and $dy/dt|_{t=0}$. There are numerous free differential equation solving packages available on-line, and I believe that Matlab also has such packages. I know that Maple and Mathematica have several such facilities.

Anyway, for given initial conditions you get a pair of differential equations that are valid only up to the point $t = t_0 > 0$ where $y(t)$ hits zero; for $t > t_0$ the solution of the differential equations no longer represent physical reality, since they would have the projectile burrowing down through the earth.

So, you can estimate the range $x(t_0)$ for any given initial conditions, but getting truly accurate results may need quite a bit of work. Of course, the range estimates are only as good as the accuracy of the numerical DE solutions, so you need to use robust and well-studied numerical DE packages. Don't try to write one for yourself; the ones in current use have been developed by teams of researchers over decades of work.

I am not an expert in this area, so I am not 100% sure that symbolic solutions are non-existent---I am only about 99.9% sure. (I have, however, carried out some numerical analyses of the type of situation you describe, and I know it is practical and not too time-consuming if using Maple, as I have done.)

Further research on your part may turn up something useful, such as special functions and the like related to your problem.

 

[ficku1]

Ok, thank you very much for your help.
I hope this will help me solve the problem.