Teaching objectives
Why can a curved slide be faster than a straight one, even when it is longer? This question obsessed Galileo, the Bernoulli brothers, and Newton. The lab starts with that broken intuition — the straight line is not the fastest — and guides secondary-school students to discover the exact answer: the cycloid. Along the way a second equally beautiful surprise appears: the tautochrone property.
What you'll learn
Students work on geometric reasoning and physical intuition: what does a curve that dives steeply at the start gain? They move from free exploration to experimental proof.
- In room 1 students choose between four slides (straight, concave, convex, "doesn't matter") and see that shape really does matter.
- In room 2 they draw their own curve freehand; the simulator measures it and compares it to the true optimum, scoring by closeness.
- Rooms 3 and 4 build the cycloid from scratch: students observe the glowing point on a bicycle wheel and identify the curve; then a slider varies the distance to the centre to show when loops appear (hypocycloid/epicycloid).
- Room 5 introduces the history of the challenge: Galileo (arc), Bernoulli (parabola), and Newton (cycloid).
- Rooms 6 and 7 confirm experimentally that the cycloid beats the parabola and the arc in a ball race.
- Rooms 8 and 9 reveal the tautochrone property: two balls released from different points on the cycloidal arc reach the bottom at exactly the same time, always.
Key mathematical ideas
- The brachistochrone (from Greek: "shortest time") is the cycloid, not the straight line nor the circular arc.
- The cycloid is the curve traced by a point on the rim of a wheel rolling without slipping.
- Varying the distance of the point from the centre produces the prolate trochoid (d > R, with loops) and the shortened trochoid (d < R, without loops).
- The cycloid is simultaneously a brachistochrone (fastest descent) and a tautochrone (descent time independent of starting point).
- Conservation of energy explains the advantage: diving steeply at the start converts potential energy into kinetic energy sooner.
Room-by-room contents
Room 1 · The ball race
The student sees four slides with different shapes (straight, concave, convex and "doesn't matter") and chooses which one the ball will reach the end of fastest. The correct answer is the convex one: the steep initial drop gives extra speed right from the start.
Student tasks
- Choose which of the four slides brings the ball to the end fastest.
- Read the explanatory feedback for each option.
Room 2 · Design the fastest curve
The student freely draws their own slide by dragging from the starting point to the finishing point. When released, the ball travels the drawn curve and its time is compared with that of the optimal curve.
Student tasks
- Draw freehand the curve you think is fastest.
- Observe the time difference compared to the optimal curve and try again if desired.
Room 3 · What curve does the wheel trace?
An animation shows a glowing point fixed to the rim of a rolling bicycle wheel. The student identifies what type of curve this point traces among four options (sinusoid, cycloid with loops, smooth arcs, cycloid).
Student tasks
- Observe the animation of the rolling wheel.
- Select the correct name of the curve traced.
Room 4 · Points on the train
The question is whether all points of a moving train travel in the same direction at the same speed. The surprising answer: points on the wheel that are below the rail move in the opposite direction to the train.
Student tasks
- Answer whether all points of the train move the same way.
- Read the clue about the wheel animation to understand why part of the wheel goes backwards.
Room 5 · The brachistochrone
Expository room: the history of the problem is told, its Greek name ("the one of shortest time") and the hypotheses of Galileo (arc of a circle), Johann Bernoulli (parabola) and Isaac Newton (cycloid). It prepares the student to check who was right.
Student tasks
- Read the historical presentation of the brachistochrone problem.
Room 6 · Explore the cycloid
A slider allows the distance d of the tracing point from the centre of the wheel to be changed. The student observes how the curve transitions from a classic cycloid (d = R) to having loops (d > R) or flattening (d < R), and answers in which case loops appear.
Student tasks
- Move the slider to see how the shape of the cycloid changes.
- Identify for which value of d loops appear in the path.
Room 7 · Which slide is faster?
Three balls compete simultaneously along a parabola, an arc of a circle and a cycloid. The student predicts who wins before the animation is launched, and confirms that the cycloid beats the other two.
Student tasks
- Predict which curve will bring the ball to the end fastest.
- Watch the animated race and confirm that the cycloid wins.
Room 8 · Experiment with the cycloid
The student places two balls at arbitrary points on the cycloidal arc and releases them at the same time. The experiment reveals that both arrive at the bottom simultaneously regardless of their starting position.
Student tasks
- Place the two balls at different positions on the arc.
- Press "Release!" and observe whether they arrive at the same time.
- Answer whether this always happens or only in specific cases.
Room 9 · The tautochrone property
The tautochrone property is formalised: two balls released from different points on the cycloid arrive at the bottom at the same time. The student chooses from three options ("the lower one first", "the upper one first", "both at the same time") and closes the lab with this astonishing result.
Student tasks
- Select in which order two balls launched from different heights on the cycloid reach the bottom.
Rooms to project
The most striking ones to show and discuss in class.