📐 IB Math AA HL IA 피드백 보고서

"[수정 예시] In basketball, players intuitively adjust their shooting angle to maximize scoring efficiency. While the ideal projectile angle in a vacuum is known to be 45°, real-world factors such as air resistance and release height complicate this relationship. This investigation aims to determine the optimal launch angle for a basketball free-throw, considering the effects of initial height and drag force. Although air resistance introduces significant complexity, a simplified model will first be developed to establish a baseline, followed by an evaluation of its limitations."

"[수정 예시 - 1단계] From the horizontal motion: x(t) = V cosθ · t → (1)
From the vertical motion: y(t) = V sinθ · t - (1/2)gt² → (2)
When the ball lands, y(t) = 0. Solving (2) for t (excluding t=0): t = (2V sinθ)/g → (3)
Substituting (3) into (1): R(θ) = V cosθ · (2V sinθ)/g = (V² · 2sinθcosθ)/g = (V² sin2θ)/g [수정 예시 - 2단계 (고급)] However, since a basketball is released from a height h (approx. 2.2m for my height), the landing condition becomes y(t) = -h, not 0. This leads to a quadratic equation: (1/2)gt² - V sinθ · t - h = 0, which modifies the final range equation significantly.
→ 이렇게 하면 'SL 수준'에서 'HL 수준'으로 점프합니다!

"[수정 예시 - 분석 확장] To rigorously prove that 45° yields the maximum range, I differentiated R(θ) with respect to θ: dR/dθ = (V²/g) · 2cos(2θ)
Setting dR/dθ = 0 to find stationary points: cos(2θ) = 0 → 2θ = 90° → θ = 45°. Furthermore, the second derivative d²R/dθ² = -(4V²/g)sin(2θ), which is negative at θ = 45°, confirming it is a local maximum. → 여기에 실제 농구선수의 슛 각도 통계(참고문헌)를 추가하면 만점에 가까워집니다.

"[수정 예시 - 결론] While the mathematical model confirms that a 45° launch angle maximizes range under ideal projectile motion, this conclusion is strictly limited to conditions without air resistance and with a zero release height. In reality, basketball shots involve a release height of approximately 2.2m and significant drag force. My simplified model therefore overestimates the optimal angle for shorter distances. [수정 예시 - 반성] If I were to extend this investigation, I would incorporate a drag force proportional to velocity (F_d = -kv) and use numerical methods (e.g., Euler's method) to simulate the trajectory. Additionally, collecting actual motion-capture data from basketball free-throws would allow me to compare my theoretical model with empirical evidence, thereby evaluating its practical validity. → 이렇게 쓰면 Criterion D에서 최고 점수(6점 만점에 5~6점)를 받을 수 있습니다.