# Projectile motion with quadratic drag and constant wind

I'm aware that there are a few similar questions already answered, but I could not find what I was looking for in any of then, so please bear with me :)

For a school project I need to find an angle ($\theta$) of maximum range ($R$) and its dependency on initial velocity $\theta(v_0)$, considering quadratic drag force and wind blowing in the direction opposite of the throw.

Drag force in $x$ direction:$F_{\textrm{drag},x}=-Kv^2\cos(\alpha)=-Kv_x(v_x^2+v_x^2)^{1 /2}$

Drag force in $y$ direction:$F_{\textrm{drag},y}=-Kv^2\sin(\alpha)=-Kv_y(v_x^2 +v_x^2)^{1/2}$

Wind force: $F_\textrm{wind}=F$

In x direction: $F_{w,x}=F\cos(\theta)$

In y direction: $F_{w,y}=F\sin(\theta)$

So the equations should look like this:

\begin{align}mv_y' &= -kv_x\cdot (v_x^2 + v_y^2)^{1/2}- F\cdot \cos\theta \\ mv_x' &= -kv_y\cdot (v_x^2 + v_y^2)^{1/2}- F\cdot \sin\theta- mg \end{align} Now I'm not sure how to solve this system of two non-linear differential equations... I'm figuring it should be done numerically, but I'm not very familiar with programming (I've only programmed a little bit in Python, but not this kind of stuff). Thank you for any answer!

• 2D quadratic drag was also considered in this Phys.SE post and links therein. – Qmechanic Sep 15 '16 at 10:15

Forget about the force $F$ of the wind as an extra term rather note the the speed of the projectile in the x-direction relative to the air will be speed of the projectile in still air (relative to the ground) $v_x(t)$ plus the speed of the wind relative to the ground $w_x$ which is a constant.

The next thing to note is that you can resolve the frictional forces into two components $K v^2_y(t)$ and $K (v^2_x(t) + w^2_x)$ and use them appropriately to find the equations of motion in the $x$ and $y$ directions.
This will mean that you will get two differential equations one in $x$ and $t$ and the other in $y$ and $t$ so solving them will not be quite as daunting?

Most Physics problems end up with one having to solve differential equations the vast majority of which do not have analytic solutions.

In Feynman - Lectures on Physics Section 9–6 Numerical solution of the equations there is a description of how one might solve a differential equation but there are better methods which you might care to research?

There are applications like Mathematica (and its baby brother WolframAlpha) which can do the job for you.

• Thank you for your answer, it is very helpful. I would just like to ask you why you only consider the x component of wind speed? Like when you resolved the frictional forces into x and y component. – nunca13 Sep 16 '16 at 5:19
• You said that the wind was in a direction opposite to the throw so I assumed it was in the minus $x$ direction. Did you mean that the wind had a downward (minus $y$) motion? – Farcher Sep 16 '16 at 5:24
• I meant it was blowing at the angle, at which the projectile was launched. I just got confused because of this misunderstanding. – nunca13 Sep 16 '16 at 6:49
• In practice I think that it would be unusual for the wind to blow in the direction you are suggesting especially if the projectile is launched from ground level. If you are allowed to do so, I would have the wind just blowing horizontally but if that is not allowed then you need an extra term in the equation of motion for the $y$ direction and note that that component of the wind will be in the opposite direction to the upward motion of the projectile but in the same direction as the projectile is coming down. – Farcher Sep 16 '16 at 7:35