Free Throw Simulation

simulation · physics

private · Python, NumPy, Matplotlib

Overview

This was a team-based final project for CIS 343: Advanced Data Analysis: Simulations at Washington & Jefferson College. In groups of five we were tasked with creating a simulation project that could be presented at the end of the semester. I presented the project idea of a free throw simulator which was selected as one of the two projects. This simulation was validated against real data that our team collected and took into account all real life forces.

Because this was a group project for a class, I am not able to share or host the source code because it is not fully owned by me and I do not have permission to distribute it. To illustrate the functionality, I have included screenshots of the interface and if you would like a live demo feel free to contact me.

The model

Our team worked to make our digital twin of a free throw shot as accurate as possible, taking into account the physics of the shot and the environment. However, due to time and computation constraints we had to make some assumptions, our biggest simplification was treating the shot as a two-dimensional problem. although this isn't fully accurate, it provides a good approximation for educational purposes. We also left out rim and backboard physics for similar reasons, this also allowed us to model our "free throw shot" with any object (like the trash can we used for our in class demo).

We identified several key factors that influence the outcome of a free throw shot, including the release angle, initial velocity, and environmental conditions. We used classic kinematic equations accounting for all of this plus drag and the magnus effect to get as accurate of a model as possible. All of the equations we used for our modeling can be seen below:

Projectile motion

Eq 1x = v t cos θ

Eq 2y = h + v t sin θ − ½ g t² taking y(0) = h, the release height

Eq 3y = −x² g2 v² cos²θ + x tan θ + h eliminating t between Eq 1 and Eq 2 — the trajectory in x alone

Eq 4v = xcos θ √g / 2x tan θ + h − H solved for the release speed that puts the ball through a target of height H at distance x

Make condition

Eq 5sin α = r2R − r r ball radius, R rim radius — the steepest entry angle that still clears, given |AB| + |BC| = |AC|

Forces on the ball

Eq 6F_D = ½ ρ v² C_D A drag, opposing the velocity vector; ρ = 1.225 kg/m³, C_D = 0.47 for a sphere, A = π r²

Eq 7C_L = 1.6 ω A / 2v lift coefficient from the spin rate ω

Eq 8F_M = C_L ρ A v ω Magnus force, acting perpendicular to the velocity — this is what backspin buys you

Integration

Eq 9a_x = F_xm a_y = F_ym − g drag contributes (−v_x/v, −v_y/v) and Magnus (−v_y/v, +v_x/v), stepped forward at Δt = 0.01 s until y ≤ 0

Validation

The most important part of any simulation is making sure you validate your output against the real world, in order to do this our team recorded 50 free throw shots and analyzed them with a video software to extract all of our parameters and get the real flight path, we then ran our simulation with these same parameters and compared the two trajectories. With this analysis we saw that our simulated path was within 1% of the real life trajectory showing that our model was accurate

The interface

Because we used this during a demo and to allow for easier testing we also created a frontend interface that takes in all initial parameters and displays the shot. Images of this interface can be seen below, the main page takes simple release inputs that allow you to take multiple shots in a row at one location while the more advanced settings allow you to edit the rim and location of the hoop you are shooting in

shot parameter panel — Shot parameters, the environment settings page, and the resulting trajectory.

environment and ball settings panel — Shot parameters, the environment settings page, and the resulting trajectory.