Analysis and calculus · Maths EE idea · Ambitious
The fastest slide: why a cycloid?
A research question to start from
Why is the cycloid the curve of fastest descent between two points, and how much faster is it than a straight line or a circular arc?
A starting point, not your question: change the case, the comparison or the limit until it is yours. The research-question builder helps you check it.
Why it works as a maths EE
A famous optimisation with a derivation and a comparison you can compute exactly.
Mathematics you would need
- Parametric curves
- Integrals for time of travel
- Energy conservation
- Introduction to the calculus of variations
Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.
One possible line of attack
- Set up the time integral for a general curve.
- Derive the cycloid (citing and explaining the Euler–Lagrange step).
- Compute descent times for several curves and compare.
Scope and difficulty
Ambitious. Ambitious; the calculus of variations must be explained, not just quoted.
Pitfalls
- Physics essay with thin mathematics.
- Times compared without exact integrals.
Where to start reading
Search a library catalogue or a university's open lecture notes for: brachistochrone derivation cycloid; time of descent comparison. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).