Question: integrating falling body motion with drag

[cooperrc]

I have a question about part 2 question 1 on homework 3. The question instructs us to integrate the equations from the fall_drag function and use those to plot the motion of the balls. I am having trouble figuring out how to integrate these equations. I assume it is talking about the differential equation from the notes:
ydot = [ v , -g+a_drag ]
, but the equation for a_drag is in terms of velocity, so the integration for this becomes quite complicated to try and get the position equation in terms of time. Am I thinking about this problem in the wrong way? Or since the objective of this question in the plot the motion, should I just be using the functions from the notes?

[cooperrc]

This is a great question, the second order ODE is,

$\ddot{x} = -g + \frac{C}{2}\dot{x}^2$

its not easy to find an equation $x(t)$ that satisfies this ODE.

Instead, we can numerically integrate with

$y_{i+1} \approx y_{i} + \dot{y}_i\delta t$

where $y = [x,\dot{x}]$ and $\dot{y}= [\dot{x},\ddot{x}]$

In Python, we need to define the ODE with a function that takes [x, v] and returns [v, a]

e.g.

def fall_drag(y):
  return np.arrray([y[1], -g + C/2*y[1]**2])