c# - Adding air drag to a golf ball trajectory equation

I'm developing a 2D golf game in VB.NET 2005, but I am stuck on how to implement air or wind drag that should affect the ball.

Already I have these equations for projectile:

• $v_0$ for the initial velocity of a golfball when hit or fired


• 
  

  Vertical and horizontal components the velocity of the golfball: $$\begin{align} v_x &= v_0 cos(\theta) \\ v_y &= v_0 sin(\theta) - gt* \end{align}$$

  
  


• 
  

  Vertical and horizontal distance of golfball: $$\begin{align} x &= v_0cos(\theta)t \\ y &= v_0sin(\theta) t - (0.5)gt^2 \end{align}$$

  
  

How do I add air drag to this equation to properly affect the velocity of the golf ball? I don't have any idea how to do it, has anyone worked with similar equations?

Answer

I'm not sure if there even exists a closed form for drag or wind, but it is quite easy to simulate in a step-wise fashion (like all the physics libraries do):

1. 
   

   set your initial condition:

   
   

   $$x, y, v_x, v_y \; (\text{for }t=0)$$

   
   


2. 
   

   update position:

   
   

   $$x = x + (v_x \times dt) \\ y = x + (v_y \times dt)$$

   
   
   

   (where dt is the time elapsed since the last update, aka delta time)

   
   


3. 
   

   calculate these velocity helpers:

   
   

   $$\begin{align} v^2 &= (v_x)^2 + (v_y)^2 \\ \lvert v \rvert &= \sqrt{v^2} \end{align}$$

   
   

   (where $\lvert v \rvert$ represents the length of $v$)

   
   


4. 
   

   calculate drag force:

   
   

   $$f_{drag} = c \times v^2$$

   
   
   

   (where c is the coefficient of friction small!)

   
   


5. 
   

   accumulate forces:

   
   

   $$\begin{align} f_x &= \left(-f_{drag} \times {v_x \over \lvert v \rvert}\right) \\ f_y &= \left(-f_{drag} \times {v_y \over \lvert v \rvert}\right) + (-g \times mass) \end{align}$$

   
   

   (where $mass$ is the mass of your golf ball)

   
   


6. 
   

   update velocity:

   
   

   $$v_x = v_x + f_x \times \frac{dt}{mass} \\ v_y = v_y + f_y \times \frac{dt}{mass}$$

   
   
   

That's basically Euler's Method for approximating those physics.

---

A bit more on how the simulation as requested in the comments:

• The initial condition $(t = 0)$ in your case is

$$\begin{align} x &= 0 \\ y &= 0 \\ v_x &= v_0 \times cos(\theta) \\ v_y &= v_0 \times sin(\theta) \end{align}$$

It's basically the same as in your basic trajectory formula where every occurrence of t is replaced by 0.

• 
  

  The kinetic energy $KE = 0.5m(V^2) $ is valid for every $t$. See $v^2$ as in (3) above.

  
  


• 
  

  The potential energy $ PE = m \times g \times y $ is also always valid.

  
  


• 
  

  If you want to get the current $(x,y)$ for a given $t_1$, what you need to do is initialize the simulation for $t = 0$ and do small dt updates until $t = t_1$

  
  



• 
  

  If you already calculated $(x,y)$ for a $t_1$ and you want to know their values for a $t_2$ where $t_1 \lt t_2$, all you need to do is calculating those small dt update steps from $t_1$ to $t_2$

  
  

Pseudo-Code:

simulate(v0, theta, t1)
  dt = 0.1
  x = 0
  y = 0
  vx = v0 * cos(theta)

  vy = v0 * sin(theta)
  for (t = 0; t < t1; t += dt)
    x += vx * dt
    y += vy * dt
    v_squared = vx * vx + vy * vy
    v_length = sqrt(v_squared)
    f_drag = c * v_squared
    f_grav = g * mass
    f_x = (-f_drag * vx / v_length)
    f_y = (-f_drag * vy / v_length) + (-f_grav)

    v_x += f_x * dt / mass
    v_y += f_y * dt / mass
  end for
  return x, y
end simulate