Maths IA · Calculus · topic 16 of 40

Pursuit curves: does the chaser ever catch up? · IB Maths IA idea: guide, sample graph and step-by-step plan

stretching! a classic in a new setting

A chaser always aims at a runner who keeps moving. The path it traces is not a straight line, and not a circle. Does it ever get there?

Difficulty●●● Stretch
KitGraphing software, a spreadsheet or Python; a field and a friend if you want data
TimeAbout 14 hours
FitsAA HL
  • ✓ you like a classic problem in a new setting
  • ✓ you enjoy algebra with a clear picture behind it
  • ✓ you are ready for a stretching HL topic

The big idea

the 'new' is where your marks are

A pursuer heads straight for a target at every moment, while the target moves along a known path. The pursuer's direction keeps changing, so its path comes from a differential equation. For a target running in a straight line, you can solve it exactly, but the classic solution needs a second-order equation and an integral beyond the course, so introduce each tool and justify it. You can then ask when, and whether, the pursuer catches up, and how that depends on the ratio of speeds. A good IA takes the classic problem somewhere new and checks it by a second method such as numerical simulation. At HL, show the substitution you use, and prove claims rather than only illustrating them.

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 →

A target that turns

Move the target in a circle, or along a path you choose, instead of a straight line.

→ The equation is harder, often with no closed form. You solve it numerically and judge accuracy and step size.
Change the speed ratio

Keep the set-up but vary the ratio of the speeds, including ratios where the target is faster.

→ The behaviour changes qualitatively: capture, no capture, or a near miss. You say exactly where the boundary lies and why.
Build it from parametric equations

Describe both bodies by position against time, then derive the pursuer's equations from 'always aims at the target'.

→ You link parametric, vector and differential-equation ideas, which is the type of connection HL rewards.
Test with real motion

Film two people, or two radio-controlled cars, and track them.

→ You compare an ideal model with messy data. Reaction delay and path wobble become real evaluation points.
Change the pursuit rule

Aim ahead of the target's position instead of straight at it, with a lead that you choose.

→ A new equation, and a question about whether the new rule catches the target sooner or at all.
Many chasers

Four pursuers starting at the corners of a square, each chasing the next.

→ Symmetry makes the problem workable. You find the path and the time until they meet, with a proof.

Your Maths IA research question: run the check

Write your own question first. Then make sure it ticks every box:

  • Names the target's path and the pursuer's rule, and says what is held fixed (the speeds, the start).
  • Says what you will find (the path, the time of capture, or the condition for capture) and for which speed ratios.
  • Says how you will check the exact solution: numerical method, simulation or data.
  • Includes something beyond the classic straight-line case, so the work is yours.
  • Is narrow enough for 12 to 20 pages and shows the key substitution and integration steps.

Get your RQ checked · S$29

The picture to draw

Pursuer and target at several moments, with the sight lines. The pursuer always heads along the dashed line. Draw yours with your own start points.

uatarget: straight line, speed upursuer: speed v,always aims at the targetdashed: sight lines at set times

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.

Aim and context~1 page
The problem, why you find it interesting, and a prediction about when capture is possible.
The model~2 pages
Coordinates, speeds and the rule 'always head for the target', written as equations with every symbol defined.
Deriving the differential equation~2.5 pages
How the rule becomes an equation for the path, with the steps shown and justified.
Solving it~3 pages
The substitution, the integration, the use of starting conditions, and the path. Checked against a case you know.
Capture~2 pages
Whether and when the pursuer reaches the target, and how that depends on the speed ratio, argued from your solution.
Second method~2 pages
A numerical simulation or real data, compared with the exact result, with step size and a number for the difference.
Conclusion~1 page
Back to the aim, with the condition for capture stated clearly.
Evaluation~2 pages, the biggest part!
Assumptions (point objects, constant speeds, instant steering), the accuracy of the simulation, what a richer model would add.
Bibliographynot counted
Sources for the classic problem and any method you borrowed.

What your Maths IA graph could look like

equal scales, or the path lies

Made-up form: a pursuer's path for two speed ratios. Yours will look different. example only

0123456789100246810121416x / m (the target runs along x = 0)y / mtarget at 0.5 of the pursuer's speedtarget at 0.7 of the pursuer's speedmade-up tracked positionsfaster target: the curve endsmuch higher up. Why?ILLUSTRATIVE · MADE-UP DATA
  • Two speed ratios on one graph, so the effect of the ratio is something the reader sees.
  • The pursuer's path against the target's line. Say where capture happens and why, from your equation, not just the picture.
  • Use equal scales on both axes so the path is not stretched (this sketch does not), and state it in a caption.
  • Tracked data with error bars shows how far the real motion is from the ideal rule.
  • Another graph to consider: distance between pursuer and target against time. It shows whether, and how fast, the gap closes.

Your game plan

simulate EARLY: it checks your algebra

Tick steps off as you go. We'll remember on this device.

0 of 10 done

  1. Learn the classic caseRead how the straight-line pursuit problem is set up. Predict when capture happens and what the ratio of speeds does.
  2. Set up the geometryChoose coordinates, put the target on its path, and write 'the pursuer heads straight for the target' as a statement about slope.
  3. Derive the equationTurn that statement into a differential equation for the path. Show each step, and check with a simple case.
  4. Solve by handFind a substitution that reduces the order, integrate, and use the starting conditions. Check limits and signs.
  5. Trial run (1 lesson)Code a simple simulation with a small time step. Does the numerical path look like your solution?
  6. Explore speed ratiosRun or solve for several ratios. Decide where the behaviour changes and show why from the algebra.
  7. Take it somewhere newChange the target's path, the pursuit rule or the number of chasers, with the same care as before.
  8. Compare and testCompare the exact result with the simulation, or with data, as a number. Check the step size and show convergence.
  9. Reflect as you goAfter each result, say what it assumes. HL: say what you proved, and what you only checked by example.
  10. Cut, check, then sendCheck that every step of the solution reads as proper notation and nothing jumps. Give your teacher one draft.

Maths you'll need

Traps to avoid in this Maths IA

  • Copying the classic solution and changing only the numbers. You need a genuinely new case or a second method.
  • A substitution that appears from nowhere. Say why it works.
  • Claiming capture from a graph that looks as if the curves meet, without showing it from the solution.
  • Setting the pursuer's speed and the target's speed in different units.
  • A simulation with a large time step, never checked against a smaller one.
  • HL: stating the capture condition without proving it.
  • Ignoring the starting condition, so the path is only one of a family of solutions.

More Maths IA ideas like this

Make sure it's yours

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.

Get your RQ checked · S$29