IA idea · Sport
What is the best angle to shoot a free throw?
Research question
Which release angle requires the least launch speed for a free throw from my release height, and how much margin for error in speed does each angle allow?
Make it yours, then agree it with your teacher.
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Why it makes a good exploration
Players are taught to shoot with an arc — but how much arc? Projectile equations give the minimum-speed angle, and the “margin for error” question makes the answer practical rather than theoretical.
The mathematics you'll need
- Projectile motion equations (explain the physics used)
- Trigonometric identities and rearranging for v²
- Differentiation to minimise launch speed
- Sensitivity: how a speed error changes where the ball lands
- Comparison with filmed shots
Where the data comes from
Measure your release height; film your own shots and estimate release angles with Tracker.
- Tracker video analysis — Free tool to track an object frame by frame in a video and export its x–y coordinates.
- Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.
A possible outline
- Set up the trajectory equations with release height and hoop distance.
- Derive v² as a function of angle.
- Find the minimum-speed angle.
- Analyse tolerance to speed errors at different angles.
- Compare with your own filmed shots and reflect on air resistance and the rim size.
Pitfalls that cost marks
- Ignoring release height.
- Treating the hoop as a point with no size.
- Quoting results without checking against real shots.
Showing personal engagement
- Film and analyse your own shots.
- Compare a tall and a short player.
- Explain why the optimum for accuracy is not the minimum-speed angle.
See Criterion C: personal engagement for what examiners look for.
Which course is it for?
| Course | Fit | Maths to lean on |
|---|---|---|
| AA SL | Good fit | Projectile motion equations (explain the physics used); Trigonometric identities and rearranging for v² |
| AA HL | Good fit | Projectile motion equations (explain the physics used); Trigonometric identities and rearranging for v² |
| AI SL | Good fit | Projectile motion equations (explain the physics used); Trigonometric identities and rearranging for v² |
| AI HL | Good fit | Projectile motion equations (explain the physics used); Trigonometric identities and rearranging for v² |
Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.
How this idea reaches the top bands
Reflection (D)
For this idea, start with the weak points: ignoring release height; and treating the hoop as a point with no size. Say how each one affects your answer.
Use of mathematics (E)
SL: projectile motion equations (explain the physics used); trigonometric identities and rearranging for v²; differentiation to minimise launch speed; sensitivity: how a speed error changes where the ball lands; comparison with filmed shots, each explained in your own words and carried out correctly.
How sport explorations reach the top bands in general: Sport IA ideas. Guides: A · B · C · D · E · self-grade sheet.
Taking it further
Include the hoop's diameter to find the range of successful angles for a given speed.
Extending it for HL
Add an HL technique the question genuinely needs: what an HL sport exploration adds.
See a complete IA, marked
How long does a skydiver take to reach terminal velocity? (AA HL) is a complete sport exploration, marked on every criterion.
Frequently asked questions
Which IB Maths course is this IA idea for?
It works for AA SL, AA HL, AI SL and AI HL. We rate it Solid: it needs some independent work beyond class examples.
What mathematics does this IA use?
Projectile motion equations (explain the physics used), trigonometric identities and rearranging for v², differentiation to minimise launch speed, sensitivity: how a speed error changes where the ball lands and comparison with filmed shots.
Where can I get data for this IA?
Measure your release height; film your own shots and estimate release angles with Tracker. Checked sources and tools: Tracker video analysis and Desmos graphing calculator.
Turn this idea into your IA
The planner opens with “What is the best angle to shoot a free throw?” filled in: the question, the maths and the data sources.
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