Durham Hackathon

How about different masses of balls?

If we want balls of various masses we need to consider which components of our simulation are affected by the mass of the ball. We know that (without friction) the weight does not affect the speed at which a ball falls. But what about the collisions?

The momentum $p$ of an object is defined as $p = m v$. For collisions, the law of conservation of momentum holds. This means that the momentum of all balls before the collision equals the momentum of all balls after the collision. Hence, in a collision between ball $i$ and ball $j$

\[m_i u_i + m_j u_j = m_i v_i + m_j v_j\]

holds, with $u_i, u_j$ and $v_i, v_j$ being the velocities before and after the collision, respectively.

Similarly, the conservation of kinetic energy

\[\frac{1}{2} m_i u_i^2 + \frac{1}{2} m_j u_j^2 = \frac{1}{2} m_i v_i^2 + \frac{1}{2} m_j v_j^2\]

holds.

In the previous steps we have implicitly assumed that all balls are equal in mass, which simply canceled out the masses. Now we need to solve the equations with the masses still in there. After manipulating the equations for a while we get

\[v_i = \frac{m_i - m_j}{m_i+m_j} u_i + \frac{2m_j}{m_i+m_j} u_j\]

and

\[v_j = \frac{2m_i}{m_i+m_j} u_j + \frac{m_j - m_i}{m_i+m_j} u_i\]

Try adding these to complete the simulation with different masses.

Add bounce:

import turtle
import random

# We will need the edges of our box, so we set them
width = 300
height = 400
window = turtle.Screen()
window.setup(width, height)
window.tracer(0)

color = ["blue",
        "yellow",
        "red",
        "darkgreen", 
        "cyan", 
        "violet",
        "magenta",
        "orange",
        "purple", 
        "navy", 
        "brown", 
        "maroon",
        "turquoise", 
        "lightgreen", 
        "green", 
        "skyblue", 
        "black", 
        "gold",
        "gray"]

N = 10 # Number of balls
balls = [] # A list to hold the balls
mass = []  # A list to hold masses of the balls

# Set up N balls and start them in random positions
for i in range(N):
    # Now the balls also have a mass
    mass.append(random.randint(1,4))

balls.append(turtle.Turtle())
    balls[i].penup()
    balls[i].shape("circle")
    balls[i].shapesize(mass[i]**0.5)
    balls[i].color(color[i%len(color)])

# Set random starting position
    balls[i].setx(random.randint(0,height / 4))
    balls[i].sety(random.randint(0,height / 4))

# Earth's gravitational constant
g = -9.81

# Timestep size
t = 0.008

# Starting velocity is now also a list, we need one velocity per ball
ux = []
uy = []
for i in range(N):
    ux.append(random.randint(-width/10,width/10))
    uy.append(0)

while True:
    for i in range(N):
        uy[i] += g*t
        balls[i].setx(ux[i]*t + balls[i].xcor())
        balls[i].sety(uy[i]*t + balls[i].ycor())

if balls[i].ycor() < -height / 2 or balls[i].ycor() > height / 2:
            uy[i] = -uy[i]
        if balls[i].xcor() < -width / 2 or balls[i].xcor() > width / 2:
            ux[i] = -ux[i]

# Check for collisions, TODO: modify to account for the relative masses
        for j in range(N):
            if i != j and \
            abs(balls[i].xcor() - balls[j].xcor()) < 10 and \
            abs(balls[i].ycor() - balls[j].ycor()) < 10:
                        uxh = ux[i]
                        uyh = uy[i]
                        ux[i] = ux[j]
                        uy[i] = uy[j]
                        ux[j] = uxh
                        uy[j] = uyh

window.update()

Fancy a hint? You can find the solution here.