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Billards & Collisions

Simulation of a ball bouncing inside a circular boundary using event detection.

You can also find this example as a standalone Python file in the GitHub repository.

The physics of this system involves a point mass moving under gravity inside a circular container of radius :math:`l`. When the ball reaches the boundary, it reflects elastically. The reflection is computed by projecting the velocity onto the normal vector at the collision point and reversing that component.

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We define the system parameters: gravity :math:`g`, the radius :math:`l` of the circular boundary, and the initial position and velocity of the ball.

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The dynamics are modeled using two Integrator blocks: one for position and one for velocity. A Constant block provides the gravitational acceleration acting on the y-component of velocity. The Scope block records the position for visualization.

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Collision Detection and Response

The collision is detected using a zero-crossing event. The detection function computes the distance from the origin minus the boundary radius:

MATHDISPLAY0ENDMATH

When this function crosses zero from below (the ball reaches the boundary), the action function is triggered. The elastic reflection is computed as:

MATHDISPLAY1ENDMATH

where MATHINLINE2ENDMATH is the outward normal at the collision point.

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Now we assemble the Simulation with all blocks, connections, and events. The gravity acts only on the y-component of velocity (channel 1). Both position components are connected to the scope for recording.

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12:43:14 - INFO - LOGGING (log: True)
12:43:14 - INFO - BLOCKS (total: 4, dynamic: 2, static: 2, eventful: 0)
12:43:14 - INFO - GRAPH (nodes: 4, edges: 3, alg. depth: 1, loop depth: 0, runtime: 0.071ms)

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12:43:14 - INFO - STARTING -> TRANSIENT (Duration: 7.00s)
12:43:14 - INFO - --------------------   1% | 0.0s<0.2s | 3180.1 it/s
12:43:14 - INFO - ####----------------  20% | 0.1s<0.2s | 3504.5 it/s
12:43:14 - INFO - ########------------  40% | 0.1s<0.3s | 3334.1 it/s
12:43:14 - INFO - ############--------  60% | 0.2s<0.1s | 3627.6 it/s
12:43:14 - INFO - ################----  80% | 0.2s<0.0s | 4135.5 it/s
12:43:14 - INFO - #################### 100% | 0.3s<--:-- | 3991.7 it/s
12:43:14 - INFO - FINISHED -> TRANSIENT (total steps: 818, successful: 778, runtime: 254.47 ms)

Let's visualize the results. First, we plot the x and y coordinates over time:

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The 2D trajectory shows the ball bouncing inside the circular boundary. We overlay the boundary circle for reference:

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