M042 · Energy and geometry

Roller-Coaster Energy Designer

Can a roller-coaster car with enough energy to reach the top of a loop still leave the track? Yes. Energy only fixes the speed at each point. Staying on the track also needs the track to keep pushing on the car, and in a loop it can do that only if the car is fast enough for the curve. A car with just enough energy to reach the top of a loop would get there at zero speed, so it peels off before the top.

Shape a circular or elliptical loop and release a particle that slides along it. Check the support force, the heat from friction and the finish speed. Record the trajectory for as long as the particle stays on the rail, and try a design challenge with three conditions.

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-conservation residual $\text{Ready}$

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 $\text{Ready}$

Shape the track, then check contact and timing

Can a roller-coaster car with enough energy to reach the top of a loop still leave the track?

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\,

A point mass rides a rail that can push but not pull. Gravity points down, and the rail supports the mass along its left-hand normal. Rolling inertia, drag and a real train's suspension are left out.

The replay ends the first time the rail stops supporting the mass or the mass stops moving forward. Flying off, rolling back and crashing afterwards are not simulated. The track's direction changes smoothly, but its curvature jumps where the pieces join.

Grey is the designed rail, and blue is the part the run covered while the model was still valid. All force arrows share one scale. The marker is schematic, but every track coordinate uses the same true scale.

The gravity estimate uses only the acquired speed and height, and no estimate is given when friction is on. Sensor noise is the noise gain times one millisecond, a tenth of a millimetre, one millimetre per second and a hundredth of a newton.

A point particle rides on a rail that can push but cannot pull. The run stops where the support vanishes or the particle stalls going forward. The curvature jumps where the track pieces join, and the hardware is drawn schematically.

Start with the essentials

Focus question: Can a roller-coaster car with enough energy to reach the top of a loop still leave the track?
One-sentence intuition: The on-track solution holds only until the particle leaves the rail; past that point it no longer describes the motion.

Core mathematical model

Track direction, normal and support force

\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 the left normal come from the designed centreline itself. A positive support force pushes along the left normal.

Speed and heat along the track

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

Kinetic friction acts only while the support force is positive. Squared speed and heat are integrated separately.

Energy balance

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

The normal force does no work while contact holds. The heat, integrated separately, makes up the difference, so both sides match.

Staying on a circular loop

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

On a frictionless circular loop, staying in contact at the top takes more energy than just reaching the top.

Estimating gravity from recorded data

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

Fit a straight line, with free intercept and slope, to the recorded heights and speeds. Friction makes this estimate invalid, so the Lab turns it off when friction is present.

Common difficulties

An idealized ball

Typical misconceptionThe ball on screen rolls, has suspension and behaves like real coaster hardware.

Better mental modelThe model is a point particle that slides. The thickness of the rail and the marker is only for drawing.

Run the experiment

01
Tell energy and contact apart
Compare Complete circular loop, Ideal contact threshold and Energy without contact.
What to observe: The blue valid part of the path ends where the track can no longer push.
02
Design the loop
Change both loop radii, the friction, the load limit and the required finish speed.
What to observe: Having enough energy alone does not pass the three-part challenge.
03
Record the valid part of the run
Acquire the trajectory, open the raw readings, then export the CSV.
What to observe: No sensor readings are made up after the particle leaves the rail or stalls.
04
Check the numerical accuracy
Double the integration intervals, then compare the refinement time difference and the separately integrated heat.
What to observe: The time difference shows how much the result changes when the steps are refined. It is not a strict error bound.