Maths IA · Calculus · topic 23 of 97
Archery: what angle hits the gold? · IB Maths IA idea: guide, sample graph and step-by-step plan
same maths as a ball, new questionAn archer aims at the inner gold ring, 12 centimetres across, 70 metres away, and the arrow leaves the bow pointing only a few degrees above the horizontal. How much can that angle wobble before the arrow misses?
- ✓ you shoot, or you know someone who does
- ✓ you like a question about hitting one exact point, not just going far
- ✓ you want to see how small an error the gold really allows
The big idea
Write the arrow's height as a function of distance, then ask a different question from the usual range problem: which launch angle makes the arrow pass through one fixed point? The condition turns into a quadratic in tan θ, so there are two answers, a low arc and a high lob. Choose between them with a reason. Then use implicit differentiation to measure how the angle must change with speed and with target height, and how small an error the gold allows. The result is sharp, so test it on real shots filmed at a short distance and use it to predict a longer one. Say what your model leaves out.
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.
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Fix the target point and solve for the launch angle that reaches it, for several speeds and distances.
→ A quadratic in tan θ with two roots. You must say what the second root means and why an archer never uses it.Differentiate the hit condition implicitly to find how the angle changes with speed and with target height.
→ A rate of change with units (degrees per metre per second). You compare it with the width of the gold.Plot the log of the angle against the log of the speed for a fixed distance.
→ A straight line with a slope you can compare with the small-angle prediction. The gap tells you what the simple formula ignores.Add a drag force that grows with speed squared and solve the motion step by step. Halve the step and see if the answer changes.
→ No neat formula any more. You compare the numerical and exact models and discuss the error of the method.Fit launch speed from shots at a short distance, then predict the angle needed at a longer one and test it.
→ The model faces a test it did not train on. A good evaluation looks at what the prediction missed and why.Track the arrow, a ball or a dart frame by frame and fit a parabola to the coordinates.
→ You estimate speed and angle with uncertainty, so the sensitivity result becomes something you can check.Your Maths IA research question: run the check
which target? how big is the gold?Write your own question first. Then make sure it ticks every box:
- Names the target: its distance, its height and the size of the zone you want to hit.
- Says what you vary (distance, speed, drag) and what you hold fixed.
- Asks for a result that needs algebra or calculus, not only substituting into a range formula.
- Says how the model will be tested against shots you filmed or measured, and what counts as agreement.
- States what the model leaves out (drag, wind, arrow wobble) and how you will judge its size.
The model
Release height, launch speed and angle, and a target a set distance away. Mark your own numbers and decide how big the gold is.
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 shooting situation, distances and arrow, with a prediction of how precise the aim has to be.
Axes and sign conventions, height as a function of distance, assumptions listed.
The quadratic in tan θ, both roots, and the reason for choosing one.
Implicit differentiation for the change in angle with speed and height. A number for the tolerance in speed and angle.
How you filmed or measured, the scale, the speed and angle estimates with uncertainty.
Calibrate on one distance, predict another, and put predicted and measured heights on one graph.
Back to the aim: which angle, how precise it must be, and how much the model could be trusted.
Drag, wind, arrow wobble, reading positions from video. Which matters most, and how could you measure it?
Sources for the model and for any archery rules or equipment data.
What your Maths IA graph could look like
tiny angles, big missesMade-up shots to show the form of a strong graph, not the answer: height of an arrow against distance for a 60 m/s launch. Yours will look different. example only
- Height against distance with units and uncertainty on both axes, so the reader sees how good the data is.
- Two angles drawn on purpose: a small change in angle moves the arrow a long way downrange.
- Compare model and points on one graph, then give a number for the typical gap.
- Say how you scaled the video: a known length in the frame, and how square-on the camera was.
- Another graph to consider: required angle against distance, one curve for each arrow speed.
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 out how archery targets and distances work. Before any calculation, guess how many degrees above the horizontal an arrow leaves the bow.
- Choose the situationPick the distance, target height and release height you can justify, and decide what size of zone counts as a hit.
- Build the flight equationsWrite height against distance, define every symbol and check the equations in a case you already know.
- Trial run (1 lesson)Film a few shots at 5 or 10 metres, or a launcher against a wall. Can you read positions off the video and keep the speed steady?
- Solve the hit conditionTurn the condition into a quadratic in tan θ, solve it, and say which root is physical and why.
- Measure the sensitivityDifferentiate implicitly, find the tolerance in speed and angle, and compare it with the size of the target zone.
- Collect and compareCalibrate the speed at one distance, predict the angle for another, and test the prediction on new shots.
- Go further, with a reasonAdd drag and solve step by step, or fit the log-log line for the angle against speed. 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 archer 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
- Parametric motion – Write and , and eliminate to get the height as a function of distance.
- Quadratic in – Use to turn the hit condition into a quadratic, solve it and interpret the two roots.
- Implicit differentiation – Differentiate the hit condition to get and , and compare the aim tolerance with the size of the gold.
- Log-log graphs – Plot against and compare the gradient with the small-angle prediction of .
- Numerical solution of an ODE – Solve the motion with quadratic drag by Euler or RK4, and halve the step to check the result.
Key equations
Syllabus links
- 2.7Quadratic equations, inequalities and the discriminant
- 3.5Unit circle definitions of sine, cosine and tangent
- 3.6Pythagorean and double-angle identities
- 5.6Standard derivatives; chain, product and quotient rules
- 5.9Kinematics
- 5.14Implicit differentiation, related rates, optimisationHL
- 5.18First-order differential equations; Euler's methodHL
- 2.10Log scales, linearising data, log-log and semi-log graphsHL
- 5.6Stationary points
- 5.9Further derivatives; chain, product, quotient rules; related ratesHL
- 5.13KinematicsHL
- 5.16Euler's method, including coupled systemsHL
Read up first
Data and sources to start from
Film shots (or a launcher) at 5 to 10 m and track the arrow frame by frame, using a measured length in the frame as the scale; use the published target-face sizes from the rules of the sport.
References to cite
- Halliday, D., Resnick, R., & Walker, J. (2018). Fundamentals of physics (11th ed.). Wiley. (Chapter on projectile motion.)
- World Archery. (2026). World Archery rules and target specifications. https://www.worldarchery.sport/
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
two roots: say which and why- Keeping only one root of the quadratic without saying why the other is not used.
- Using the small-angle shortcut sin 2θ ≈ 2θ without saying when it stops being safe.
- Taking launch speed from a poster or a catalogue when your own bow or launcher is different. Measure it.
- Reading positions from video without a scale, or with the camera at an angle to the flight.
- Calling the model good because one shot hit. Say how many shots, and what the typical error was.
- Making drag the whole evaluation, with no estimate of how big it is.
- Copying the setting of an existing projectile IA without a new question to answer.
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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.