Maths IA · Calculus · topic 26 of 97
Netball and basketball: which shot is most forgiving? · IB Maths IA idea: guide, sample graph and step-by-step plan
"best" has to be defined firstEvery coach says shoot with more arc. A steeper shot has a bigger hole to aim at, but it needs more speed. So where does the best angle really sit?
- ✓ you play netball or basketball and want to know why coaches say "arc it"
- ✓ you like questions where the best answer depends on how you measure "best"
- ✓ you are happy to combine calculus, trigonometry and a little probability
The big idea
From a fixed release height, a ball must reach a ring 3.05 metres up. For each release angle there is exactly one speed that hits the centre. Write that speed as a function of the angle and find the angle that needs the least speed: the double-angle identities make this clean. Then ask the better question. The ball has a width, the ring has a width, and your arm is never exact. How far can speed and angle be off before the ball hits the rim? That margin changes with the angle, and it changes differently for speed errors and for angle errors. The most forgiving angle is not the minimum-speed angle. Show where they differ, and say what assumptions decide it.
Make it yours: pick one twist
Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.
Swipe for more twists →
For a fixed release height and distance, write the speed needed as a function of the angle and minimise it.
→ A calculus problem with a neat answer: the angle is 45° plus half the angle the ring makes with the release point. Prove it two ways.Work out how far the ball's centre can be off at the ring and still go in, using the ring width and the ball width.
→ The margin depends on the entry angle. You turn it into an allowed error in speed and in angle with derivatives.Compare the allowed error in speed with the allowed error in angle, as percentages or in matching units.
→ The answer changes with the release angle. You find the angle where the two balance and say what it means for a shooter.Treat speed and angle errors as normal, with standard deviations from filmed shots, and find the chance of scoring.
→ The best angle now depends on how consistent the shooter is. You compare a steady and a shaky shooter.Use the netball ring (narrower, no backboard) and the basketball ring, with their own ball sizes.
→ The minimum entry angle changes, and so does the best shot. You compare the two sports with numbers.Film free throws or netball shots, estimate the release angle and entry angle, and compare them with your model.
→ Real players do not use the model's best angle. Your evaluation asks why: spin, the backboard, comfort.Your Maths IA research question: run the check
best for what? say how you measure itWrite your own question first. Then make sure it ticks every box:
- Fixes the situation: release height, distance and the ring and ball sizes you will use.
- Says how "best" is measured: least speed, widest margin, or chance of scoring.
- Asks something that needs differentiation or a distribution, not only a trajectory formula.
- Says how the model is compared with filmed or published shots.
- States what the model leaves out (spin, backboard, air drag) and how you will judge its size.
The model
A release point, a ring and a ball with a width. Mark your own heights and distance, and think about what "going in" needs.
How your Maths IA could be laid out
The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.
The sport, the shot, and a prediction: is the best angle the one that needs least speed?
Axes and symbols, speed needed as a function of angle, assumptions listed.
A derivative or a trigonometric identity, the result checked in a case you know.
The entry angle, the size of the hole the ball sees, and the allowed error at the ring.
Derivatives turn the margin into allowed errors. Graph them against the release angle.
How you filmed, the scale, release speed and angle with uncertainty, and the entry angle you saw.
Back to the aim: which angle is most forgiving, for whom, and by how much.
Spin, backboard, air drag, a point-ball model. Which matters most, and how could you test it?
Sources for the model and for ring and ball sizes.
What your Maths IA graph could look like
flat at the bottom: so what?Made-up numbers to show the form of a strong graph, not the answer: speed needed against release angle for a ring 0.85 m above the release point, 4.19 m away. Yours will look different. example only
- The curve has a minimum. Find it by calculus, then mark it on the graph.
- Error bars on both axes: you read the angle off video, so it has an uncertainty.
- The curve is flat near the minimum. Say what that means for how much angle error a shot allows.
- Plot your measured shots on top, and give a number for how far they sit from the model.
- Another graph to consider: allowed error in angle and in speed against release angle, on one set of axes.
The maths in action
worked through, not sketched!The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Read and predictFind the ring height, ring width and ball size for your sport. Predict which release angle is most forgiving, and why.
- Choose the situationPick a release height and a shooting distance you can justify, such as a free-throw line or the edge of a shooting circle.
- Build the flight equationsWrite position against time, define every symbol and find the speed needed for each angle.
- Trial run (1 lesson)Film a few shots from the side. Can you read the release angle and the ball's path? Is the camera square-on?
- Find the least-speed angleMinimise the speed by calculus, then confirm the result with a double-angle identity or a graph.
- Measure the marginWork out the entry angle, the width of the hole the ball sees, and the allowed error in speed and in angle.
- Collect and compareFilm shots from the same spot, estimate release speed and angle with uncertainty, and compare with the model.
- Go further, with a reasonModel release errors as normal and find the chance of scoring against angle. HL: this is where the rigour for criterion E comes from.
- Reflect as you goAfter each result, write one line on what it means for a real shooter and what it assumes. These lines become your evaluation.
- Cut, check, then sendCheck units, uncertainties and that the conclusion answers your aim. Then send your teacher one draft.
The maths you'll use
- Trajectory equations – Find the speed needed to reach the ring as a function of the release angle, from and .
- Double-angle identities – Rewrite the denominator as and write it as one sine to find the least-speed angle .
- Optimisation – Minimise by differentiation and prove it is a minimum, then check the result with the identity.
- Partial rates of change – Differentiate the horizontal distance at ring height with respect to speed and angle to turn the margin into allowed errors.
- Normal distribution – Combine independent speed and angle errors and find for each release angle.
Key equations
Syllabus links
- 3.5Unit circle definitions of sine, cosine and tangent
- 3.6Pythagorean and double-angle identities
- 3.10Compound angle identitiesHL
- 4.9Normal distribution
- 5.6Standard derivatives; chain, product and quotient rules
- 5.8Maxima, minima, inflexion and optimisation
- 5.14Implicit differentiation, related rates, optimisationHL
- 5.6Stationary points
- 5.9Further derivatives; chain, product, quotient rules; related ratesHL
- 4.9Normal distribution
Read up first
Data and sources to start from
Film free throws or netball shots from the side with a metre stick in the frame, and track the ball for release angle, release speed and entry angle; ring and ball sizes come from the official rules.
References to cite
- Brancazio, P. J. (1981). Physics of basketball. American Journal of Physics, 49(4), 356-365.
- Halliday, D., Resnick, R., & Walker, J. (2018). Fundamentals of physics (11th ed.). Wiley. (Chapter on projectile motion.)
- International Netball Federation. (2026). Netball rules. https://netball.sport/
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
a ball has a width- Treating the ball as a point when the whole question is about how wide the ball and the ring are.
- Quoting 45° as the best angle. With the ring above your hand it is higher, and you have to show why.
- Mixing units in the tolerances: compare errors as percentages, or convert the angle to radians first.
- Assuming the errors are normal without checking them against your own shots.
- Ignoring the backboard and spin in basketball, without saying how much they could change the result.
- Calling a model good because it matches one made shot.
More Maths IA ideas like this
Best throwing angle when landing height differs●●○
Archery: what angle hits the gold?●●○
Goals and tries: are they random?●●○
This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.