---
title: "Roller-Coaster Energy Designer Interactive Lab | Sandbox Physics"
description: "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."
canonical_url: "https://sandboxphysics.com/labs/coaster-energy-designer/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/coaster-energy-designer/"
last_modified: "2026-10-27"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

Physics tutorial

## Energy, curvature and unilateral support

**Background** The rail fixes the shape of the path, and gravity supplies the energy. How hard the track pushes and how long the trip takes depend on the whole path.

**Why it matters** The track you see, the motion and the recorded readings all come from the same model quantities, so you can compare them directly.

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 misconception** The ball on screen rolls, has suspension and behaves like real coaster hardware.

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

### Run the experiment

1. 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.

2. 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.

3. 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.

4. 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.

Physical conclusion

Check that the track can still push on the particle, not just that the energy is sufficient.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)