1. The bouncing ball

Hi all!

"The bouncing ball" consists basically in a simple red ball bouncing up and down. Also I've added a graphing function to plot the position over time where you can see the lost energy at every bounce.

Picture: 

Description:

Involved physics: 1D kinematics, differential equations (without solving them).

Needed python libraries: visual and visual.graph

Parameters: you can change the initial velocity, high, gravity, the elasticity of the bounce and friction with air.

Explaining the code: 

The following lines are used to import the two needed libraries.

#PROGRAM DEPENDENCES:
from visual import *
from visual.graph import *

-

Next step is the display were you can set up the size of the window, the title, if you want python to auto-scale the scene among others. 

#SCENE DISPLAY (FROM VPYTHON)

scene.width = 700
scene.height = 600
scene.title = 'BOUNCING BALL'
scene.autoscale=False
scene.fullscreen = False
scene.center = vector(0, 10, 10)

-

The variables' values can be changed because the equations are written as a function of them.

#VARIABLES

t_max = 20 #program timeout
g = -9.81 #aceleration of gravity
y = 10 #initial high of the ball
v = 10 #initial velocity of the ball
dt = 0.006 #increase of time for each iteration
e = 0.9 #coef. lost of energy in each bounce
k = 0 #friction with air
t = 0 #initial time

-

Here defining floor as a box and ball as a sphere (a vpython's function) for drawing them in the scene. Next step, the position of the ball is updated within a while loop.

#DRAWING THE OBJECTS

floor = box(height = 0.2, width = 30, length = 30, color =

 color.white, material = materials.wood)

ball = sphere(pos = vector(0, y, 0), radius = 1, color =

 color.red) 

-

Before entering the while, we define the display of the curve and the plot's interface.

#GRAPH
f1 = gdisplay(x=0, y=0, width=600, height=600,
          title='MOVEMENT (y vs. t)', xtitle="t", ytitle='y',
          foreground=color.black, background=color.white,
          xmax=t_max, xmin=0, ymax=15, ymin=0)
f1 = gcurve(color=color.black) #The graphics curve

-

The next step is to create a loop for updating the position of the ball. Here are used differential equations to update incrementally the position.

The idea is to start the loop with a value of dt, that causes a sequence of small increments in velocity and position. Doing that, you are drawing successive pictures of the ball, floor that produces a movement effect.

while t < t_max: #main loop
    rate (200) #loops/sec
    #DIFERENTIAL EQUATIONS
    dv = g*dt - k*v*dt
    v = v + dv
    dy = v*dt
    y = y + dy
    t = t + dt

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Now comes a crutial part, the bounce. It's obvius that the ball has to bounce when it has the floor's position, then the idea is to invert the velocity upwards. The problem is that our program has an error in position given by the magnitude of dt. This small error forbids us to put a simple if like: "if y == ball.radius:", because it's not sure that the ball will reach this exact point. So we are forced to put the next condition: "if y < ball.radius:" . But now we have another problem, the ball renters the if at the next iteration because it haven't yet reached the needed high to be upper than ball.radius. That's because if you apply some lose of energy the bounce isn't symmetric. This produces a "bug" where the ball inverts his velocity several times and can bounce downwards. The solution is to set up the ball's position to the ball.radius high after entering the if function, so avoiding this "bug".

    #Bounce conditions
    if  y < ball.radius:
        v = -v*e
        y = ball.radius

    #Scene updating
    ball.y = y

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Finally you have to update the ball.y and draw this point into the graph:

    #Drawing in the plot
    f1.plot(pos = (t , y))
  
print "...FINISH!"
print "www.pythonforphysics.blogspot.com"

That's all! Don't forget to comment if you enjoyed! :)

*You can download the program from the library or from on this link