General Form Equation for the Most Energy-Efficient Basketball Shot
Bibliographic record
Abstract
Although ball shooting is one of the fundamental skills of basketball, it is energy intensive and exhausts players quickly. Because of this, effective shooters will have less time to play on the court, which can be the difference between winning and losing for basketball teams. In the National Basketball League (NBA), pride, expectations, and millions of dollars in investments are at stake, making winning all the more critical. Improving basketball shooting accuracy may provide the most robust path forward to increasing team-specific wins. So, to boost shooter performance, we derived an equation that calculates the optimal velocity to shoot a basketball from every position on the court. This ‘optimal’ velocity minimizes a shooter’s energy expenditure so players can conserve their energy and stay effective in the game longer, boosting their team’s chances of winning. We used Newton’s second law and kinematics, vector algebra, and calculus to derive the optimal velocity equation and ultimately implemented it in Python for public use. We concluded that the optimal angle for mid-range shots (3 to 5 meters) is 55 to 51 degrees and 49 to 47 for three-point shots.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".