M042 · Energy and geometry

Roller-Coaster Energy Designer

Shape a circular or elliptical loop, release a sliding particle, and audit support, friction heat and finish speed. Acquire the physically valid trajectory and test a three-part design challenge.

Selected station time $\text{Ready}$

Selected speed $\text{Ready}$

Selected support over weight $\text{Ready}$

Peak support over weight $\text{Ready}$

Kinetic energy $\text{Ready}$

Potential energy $\text{Ready}$

Rail friction heat $\text{Ready}$

Energy closure $\text{Ready}$

Valid-prefix end time $\text{Ready}$

Refinement time difference $\text{Ready}$

Gravity from raw speed and height $\text{Ready}$

Contact, load and finish challenge $\text{Ready}$

Contact state $\text{Ready}$

Acquired stations $\text{Ready}$

GEOMETRY / CONTACT / TIMING

Can enough energy still lose contact?

Release height

H

6.00\,m

Horizontal loop radius

R

2.00\,m

Vertical to horizontal radius

b/R

1.00\,

Initial downward speed

v_0

0.00\,\mathrm{m\,s^{-1}}

Particle mass

m

1.00\,kg

Kinetic rail friction

\mu

0.00\,

Peak support load limit

G_{max}

7.50\,

Minimum finish speed

v_{min}

2.00\,\mathrm{m\,s^{-1}}

Integration intervals per piece

N

160.00\,

Replay time fraction

t/T_{max}

0.42\,

Sensor noise gain

\eta

0.00\,

Point mass on a one-sided rail. Gravity points down; the left normal defines support. No rolling inertia, drag or real train suspension.

Replay ends at the first detected loss of support or forward stall. Subsequent flight, reversal and impact are omitted. Geometry is tangent-continuous; curvature jumps at the piece joins.

Grey is the designed rail; blue is the valid prefix. Force arrows share one gain. The marker is schematic; all track coordinates use one physical scale.

Gravity estimation uses only acquired speed and height, and is withheld with friction. Sensor noise gain scales milliseconds, tenths of a millimetre, millimetres per second and hundredths of a newton.

One-sided point-particle contact model; stops at support loss or forward stall. Curvature jumps and schematic hardware are declared.

Start with the essentials

Focus question: Can enough energy still lose contact?
One-sentence intuition: Follow the contact-valid prefix; never extend the constrained solution beyond detachment.

Core mathematical model

Geometry and support

\begin{aligned}\mathbf t&=\frac{d\mathbf r}{ds}\\\mathbf n&=(-t_y,t_x)\\N/m&=v^2\kappa+g\,t_x\end{aligned}

The signed curvature and left normal use the actual designed centreline. Positive support pushes toward the left normal.

Tangential dynamics

\begin{aligned}\frac{d(v^2)}{ds}&=-2g\frac{dy}{ds}-2\mu\frac{N}{m}\\\frac{dQ}{ds}&=\mu N\end{aligned}

Only positive support supplies kinetic friction. Squared speed and heat are integrated independently.

Energy ledger

\frac12mv^2+mgy+Q=\frac12mv_0^2+mgH

Normal support does no work while contact holds. The separately integrated heat closes the ledger.

Circular loop threshold

\begin{aligned}N_{top}/m&=v_{top}^2/R-g\\v_{bottom}^2&\ge5gR\end{aligned}

For a frictionless circular loop, the contact condition is stronger than the energy needed merely to reach its top.

Raw-data gravity estimate

v_i^2=\alpha-2g\,y_i

Fit a free intercept and slope to acquired height and speed. Friction invalidates this inference and disables it.

Common difficulties

Idealized apparatus

Typical misconceptionA rendered ball includes rolling, suspension and real hardware dynamics.

Better mental modelThis is a sliding point-particle model. Rail and marker thickness are schematic.

Run the experiment

01
Separate energy and contact
Compare Complete circular loop, Ideal contact threshold and Energy without contact.
What to observe: The valid blue prefix ends where the track can no longer push.
02
Design the loop
Change both loop radii, friction, load limit and required finish speed.
What to observe: Meeting the energy budget alone does not pass the three-part challenge.
03
Acquire a physical prefix
Acquire the trajectory, open raw readings, then export CSV.
What to observe: No sensor records are fabricated after contact loss or stall.
04
Audit numerical agreement
Double the intervals and compare the time refinement difference and independent heat ledger.
What to observe: The difference is a refinement diagnostic, not a rigorous error bound.