Colliding Blocks

This Python code computes the kinematics data (linear momentum and kinetic energy) for the 2D collission between 2 blocks and a rigid wall taking into account the coefficient of restitution but ignoring the friction.

🔰 How does it work?

• The code uses VPython library to create the 2D-3D interactive environment and the physics interactions.

• Letter A is set for the red block and the letter B is set for the green block.

• User has to enter the following parameters

  • Blocks' mass.
  • Initial velocity of the blocks (in the main case the red block doesn't have initial velocity).
  • Coefficient of restitution between the red and the green block.
  • Coefficient of restitution between the red block and the rigid wall.

• After each collision the kinematic data is computed using the following equations (Notation: 1-Before collision, 2-After collision):

  • Conservation of momentum: $$m_A (v_A)_1 + m_B (v_B)_1 = m_A (v_A)_2 + m_B (v_B)_2$$

  • Conservation of energy: $$\frac{1}{2} m_A {(v_A)_1}^2 + \frac{1}{2} m_B {(v_B)_1}^2 = \frac{1}{2} m_A {(v_A)_2}^2 + \frac{1}{2} m_B {(v_B)_2}^2 $$

  • Coefficien of restitution between blocks: $$e = \frac{(v_B)_2 - (v_A)_2}{(v_B)_1 - (v_A)_1}$$

  • Coefficient of restitution between the red block and the rigid wall: $$e = -\frac{(v_A)_2}{(v_A)_1}$$

• Kinematic data for both blocks is computed for each step of time during the simulation:

  • Linear momentum
  
  • Kinetic energy
  

• There is an special set of parameters which could be used to calculated the digits of Pi $(\pi)$.

  • All the coefficients of restitution muts be equal to 1 (perfectly elastic)
  • Mass' A must be equal to 1.
  • Mass' B bust be equal to $100^{n-1}$, where $n$ is the number of digits of Pi computed based on the number of collision between all the bodies during the simulation. For example:

    • If $n = 1$ $(m_B = 1 kg)$, the number of collision would be $3$

    • If $n = 2$ $(m_B = 100 kg)$, the number of collision would be $31$

    • If $n = 3$ $(m_B = 10000 kg)$, the number of collision would be $314$

• An explanation of this peculiar phenomenon is described by G.Galperin with a lot of math, but there is an incredible and illustrative explanation in the 3Blue1Browm Youtube Channel

🔶 What is next?

• Increase the number of blocks and rigid walls.