Maths IA · Calculus · topic 35 of 97

The fastest slide: straight line, circle or cycloid? · IB Maths IA idea: guide, sample graph and step-by-step plan

The straight line is the shortest way down a slide. It is not the fastest. You will build the time of a descent as an integral and find out how much time a curve can save.

• ✓ you like a result that goes against your first instinct
• ✓ you are happy to build an integral and then calculate it numerically
• ✓ you enjoy a problem with a famous history behind it

The big idea

A ball released from rest has a speed that depends only on how far it has dropped. So the time to slide down any curve is a sum of tiny path lengths, each divided by the speed at that depth. That sum is an integral, and you can compute it for a straight line, a circular arc, a parabola or a cycloid between the same two points. Some of these have exact answers, and some need numerical integration. The famous result is that the cycloid is quickest, but the calculus of variations behind it is beyond the course, so you quote it and test it. The marks come from testing it yourself: compare your computed times with each other, and with timing a real ramp, and explain every gap.

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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Your Maths IA research question: run the check

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

• Names the two end points of the slide with their distances, and what is assumed about friction.
• Names the curves being compared and says which ones have exact times and which are computed numerically.
• Says how the integral will be checked, such as against the straight-line case or a timed ramp.
• Says what quantity is optimised (time of descent) over which parameter.
• Is narrow enough for 12 to 20 pages and does not try to derive the calculus of variations.

The model

A bead slides from rest from the start to the finish. Its speed depends only on its depth, so the time is a sum over tiny pieces of path. Draw your own start, finish and two curves.

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.

What your Maths IA graph could look like

Made-up numbers to show the form of a strong graph, not the answer: descent time against the shape of the arc, with a reference time. Yours will look different. example only

• Time against a shape parameter, with an exact reference drawn as a flat line to compare with.
• Axis labels give units: time in seconds, bulge in metres.
• Mark the minimum and the reference, and give the gap as a percentage.
• Plot your own numerical integrals as points, to show the method agrees with the curve.
• Another graph to consider: the shapes of all the paths on one set of axes with the y-axis pointing down.

The maths in action

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.

How much time does the curve save?

Rank the paths on one number: time of descent from rest. Work it out for your own end points and explain why the gaps are not equal.

Your game plan

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

0 of 10 done

1. Read and predictFind out what the brachistochrone problem asks. Before calculating, guess which of a line, an arc and a dip-then-flat path is quickest.
2. Choose the end pointsPick a start and finish with real distances, such as a playground slide or a skate ramp, and decide whether you ignore friction.
3. Speed from energyDerive the speed at depth y, then build the time as an integral. Define every symbol.
4. Trial run (1 lesson)Do the straight line two ways, by the integral and by constant acceleration. They must agree before you go on.
5. Build the arcsChoose a family of curves through your two points with one parameter. Write each as parametric equations so the integral is easy to set up.
6. Compute and searchCompute each time numerically, check it on the line, then search for the parameter with the least time. Show the method's convergence.
7. Meet the cycloidWrite it in parametric form, find the angle that reaches your finish point and evaluate the exact time. Quote the optimality result and say where it comes from.
8. Go further, with a reasonTime a real ramp, add friction or move the finish point. Say what question the extra step answers. HL: this is where the rigour for criterion E comes from.
9. Reflect as you goAfter every result, write one line on what it means for a real slide and what it assumes. These lines become your evaluation.
10. Cut, check, then sendCheck units everywhere, every graph labelled, every symbol defined once and the conclusion answers your aim. Then send your teacher one draft.

The maths you'll use

• Speed from energy – Use to write the speed as a function of depth, then form .
• Arc length element – Write or its parametric form so the time becomes an integral in one variable.
• Integration – Evaluate the straight-line time exactly, and the cycloid time exactly using its parametric form.
• Numerical integration – Compute the time along a circular arc or power curve with the trapezium rule or software, handling the start where speed is zero.
• Optimisation – Treat time as a function of one shape parameter and find its minimum with a graph or numerically.
• Parametric equations – Describe a cycloid by , and find for the finish point.

Key equations

Syllabus links

• 3.13D distance, volume and surface area
• 5.3Differentiating powers of x
• 5.8Maxima, minima, inflexion and optimisation
• 5.11Definite integrals and areas between curves
• 5.15Further derivatives and integrals; partial fractionsHL
• 5.16Integration by substitution and by partsHL

• 5.5Integration as anti-differentiation; areas
• 5.8Trapezoidal rule
• 5.9Further derivatives; chain, product, quotient rules; related ratesHL
• 5.13KinematicsHL

Read up first

• Integration (AA HL) →
• Differentiation (AA HL) →
• Approximation and error (AI HL) →

Data and sources to start from

Compute descent times from the model, then time a ball on ramps of different shapes with slow-motion video, and compare.

References to cite

1. Bernoulli, J. (1696). Problema novum ad cuius solutionem mathematici invitantur. Acta Eruditorum.
2. Stewart, J. (2015). Calculus: Early transcendentals (8th ed.). Cengage Learning.
3. Gelfand, I. M., & Fomin, S. V. (2000). Calculus of variations. Dover Publications.

Cite what you actually use, in your own format, and say where each number came from.

Traps to avoid in this Maths IA

• Assuming the shortest path is the fastest. Speed depends on depth, so a longer path can win.
• Measuring depth upwards in the speed formula, which makes the speed imaginary or the sign wrong.
• Integrating a path that starts with zero speed without noticing the integrand is infinite at the start. Say how you handled it.
• Quoting the cycloid result without checking it numerically against your own curves.
• Comparing a rolling ball with the sliding-bead model without allowing for the energy that goes into rotation.
• Using a finish point that makes the cycloid dip below the finish, and not saying so.

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