Maths IA · Calculus · topic 12 of 40
Best throwing angle when landing height differs · IB Maths IA idea: guide, sample graph and step-by-step plan
the textbook case is just ONE of manyEveryone knows the best angle for a long throw. But the pitch is rarely flat: shot put, free throw, throwing off a wall. So does the rule survive?
- ✓ you like sport and want to model it
- ✓ you can film throws and read positions off video
- ✓ you want a clean model you build yourself
The big idea
derive it, don't quote itWrite the position of a thrown ball as a pair of equations in time. Use them to find the range for any launch angle, launch speed and height difference. Then ask which angle gives the greatest range, and how that angle depends on the heights. The textbook angle belongs to one special case, so you have to derive the general case yourself and justify it rather than quote it. Then test it against throws you filmed, and say honestly where the model is too simple.
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 →
Throw up onto a platform or into a basket, so the landing point is above the launch point.
→ The same model, with the height difference changing sign. Check that your equations still work and that the throw can reach the target at all.Add a drag force that depends on speed and solve numerically (Euler or a spreadsheet).
→ No neat formula for the best angle any more. You compare numerical and exact results and discuss step size.Choose the angle that needs the least launch speed to hit a target point, or gives the longest flight time.
→ A different function to differentiate, and possibly a different best angle. You compare the criteria.Track the ball frame by frame and fit a parabola to the coordinates.
→ You estimate launch speed and angle from data, with uncertainty, then check the model's prediction of the range.Free throws in basketball, where the ball must enter the hoop at a steep enough angle, or the shot put.
→ Constraints appear (the hoop's size, the release height), so the best angle is not just the one for greatest range.Differentiate the range with respect to angle, solve for the angle, and show it is a maximum, for general speed and height.
→ An algebra-heavy result that you must prove rather than show by example. This is where the HL-level reasoning comes from.Your Maths IA research question: run the check
Write your own question first. Then make sure it ticks every box:
- Names the quantity you want to be greatest or least (range? reaching a target?) and what you vary (angle, height).
- Says what is held fixed (launch speed, which way the ball lands) and why that is realistic.
- States the model's assumptions (no air resistance? point mass?) and how you will test them.
- Says whether you compare with data you filmed, a published sport measurement, or another model.
- Is narrow enough for 12 to 20 pages and needs real algebra, not only substitution.
The picture to draw
A ball thrown from a raised point. Label launch speed, angle, launch height and range, and say where you measure each from.
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 situation you chose and why it matters, with a prediction of what happens to the best angle before you calculate.
Axes, origin and sign conventions. Position as functions of time. Assumptions listed.
Flight time and range as functions of angle, with the quadratic solved and the right root chosen for a reason.
A derivative or another justified method, the general result, and a check that it is a maximum. Test it on the level-ground case.
How you filmed, how you read coordinates, how you got speed and angle, with a table and uncertainties.
Predicted and measured range for each angle on one graph, with a number for the difference.
Back to the aim: how and why the best angle depends on the heights.
Air resistance, release height, spin, reading positions off video. Which one matters most, and how could you test it?
Sources for the model and for any sports data.
What your Maths IA graph could look like
how flat is the top?Made-up throws, to show the form of a strong graph, not the answer. Yours will look different. example only
- Both models drawn on one graph, so the effect of the height difference is visible, not just stated.
- Error bars in both directions: you set the angle by eye, so its uncertainty matters.
- Notice how flat the top of the curve is. Say what that means for how precisely the best angle can be found.
- Check the units and that your launch speed is the same for every throw. If it is not, the comparison is unfair.
- Another graph to consider: best angle against launch height, for several heights.
Your game plan
film a trial first!Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Predict firstWrite down what you expect to happen to the best angle when the ball lands lower, and when it lands higher. Say why.
- Build the equationsChoose axes and write the horizontal and vertical position against time. Check them against a simple case you already know.
- Solve for the rangeFind the time of flight, work out which root of the quadratic is physical, and write the range as a function of angle.
- Optimise and justifyDifferentiate, solve, and show the result is a maximum. If you simplify, say what you gave up.
- Trial run (1 lesson)Film a few throws. Can you keep the speed the same? Is the camera square-on? Can you read the coordinates cleanly?
- Collect the dataFilm throws at several angles from at least two heights. Scale each video using a known length in the frame.
- Extract and processRead positions, estimate speed and angle with uncertainty, and compare the measured and predicted range.
- Test the assumptionsChoose one assumption to stress (air resistance is the usual one) and see how much it changes the result.
- Reflect as you goAfter each result, ask what it means for the real throw. These notes become your evaluation.
- Cut, check, then sendCheck notation, axes and units, and that the conclusion answers your aim. Give your teacher one draft.
Maths you'll need
Traps to avoid in this Maths IA
- Quoting 45° as the answer: it belongs to a special case, and your whole point is that the case has changed.
- Taking the wrong root of the quadratic for flight time without saying why.
- Different launch speeds in different throws, then comparing the ranges as if they were the same.
- Reading positions from video without a scale or without allowing for the camera's angle.
- Putting the calculator or spreadsheet formula in the write-up instead of proper notation.
- HL: a derivation that stays at standard SL techniques. Say what is new or general in yours.
- 'Air resistance' as the only evaluation point, with no estimate of how big it is.
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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.