---
title: "How Orbits Take Shape Interactive Lab | Sandbox Physics"
description: "How can one force law draw three different conic sections? The starting position and velocity fix the energy and angular momentum, and those choose the curve."
canonical_url: "https://sandboxphysics.com/labs/orbit/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/orbit/"
last_modified: "2026-10-27"
---

Gravity · Kepler orbits

# Orbital Launchpad

**How can one force law draw three different conic sections?** The starting position and velocity fix the energy and angular momentum, and those choose the curve. Negative energy gives a bound ellipse or circle, zero gives a parabola, and positive energy gives a hyperbola that escapes. For a given energy, angular momentum sets how stretched the orbit is and how close it comes.

Launch a body into a circle, an ellipse or an escape path. Rotate the view and replay any moment. Compare the computed path with the exact conic section, then look at the areas swept in equal times and at the recorded energy.

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

The scene, the readouts and the CSV file all come from the same recorded trajectory. All quantities are in normalized units, and the bodies are drawn larger than scale.

Data from the integrated trajectory

## Does the orbit keep its energy and angular momentum?

Recorded time $t=0.00$

Solid paths are integrated samples; the dashed conic is calculated independently from the initial position and velocity. The coloured sectors start at launch and halfway through the record. Both span half a normalized time unit, shortened equally for a brief record.

$$
r_{\mathrm p}=\frac{h^2}{\mu(1+e)},\qquad \frac{\mathrm d A}{\mathrm d t}=\frac{h}{2}
$$

Time, distance and mass use normalized units. The plane and rings are coordinate guides. Display size does not represent a physical radius. Energy errors are divided by the magnitude of the initial potential energy; they remain meaningful at zero total energy. Replay never recomputes the trajectory.

[Add a third body →](https://sandboxphysics.com/labs/three-body/) [NASA: gravity and orbital motion](https://science.nasa.gov/learn/basics-of-space-flight/chapter3-4/)

Physics tutorial

## Orbits: read conic sections through energy

**Background** Gravity weakens with the square of the distance and always points toward the central body. Depending on the starting position and velocity, the same force traces a circle, an ellipse, a parabola or a hyperbola.

**Why it matters** The same reasoning predicts satellite maneuvers, planetary orbits, and escape without first solving the full trajectory.

Start with the essentials

- **Focus question**: From position and velocity alone, how can we tell whether a body returns or escapes?
- **One-sentence intuition**: The energy per unit mass decides whether the body stays bound; the angular momentum sets the shape of the orbit and its periapsis, the point of closest approach.

### Core mathematical model

#### Central gravity

$$
\ddot{\mathbf r}=-\frac{GM}{r^3}\mathbf r
$$

Acceleration always points inward and its magnitude falls as $1/r^2$.

#### Specific mechanical energy

$$
\varepsilon=\frac{v^2}{2}-\frac{GM}{r}
$$

$\varepsilon<0$ is a bound ellipse, $\varepsilon=0$ is the parabolic boundary, and $\varepsilon>0$ means escape on a hyperbola.

### Common difficulties

#### Gravity does not guarantee return

**Typical misconception** A body must return because gravity never disappears.

**Better mental model** Gravity weakens with distance. If the total energy is positive, the body escapes even though it keeps slowing down.

#### Circular motion still accelerates

**Typical misconception** Constant speed means zero acceleration.

**Better mental model** The direction of the velocity changes all the time, and that takes a centripetal acceleration pointing inward.

### Run the experiment

1. 01

   #### Identify a circular orbit

   Choose Circular and compare velocity with gravity.

   **What to observe:** The two arrows are nearly perpendicular and the radius stays steady.

2. 02

   #### Change the energy

   Keep the launch radius fixed, raise the speed step by step and launch a new record each time.

   **What to observe:** The ellipse stretches longer and approaches the escape boundary.

3. 03

   #### Cross the escape threshold

   Choose Escape and inspect the classification, predicted periapsis and conservation errors.

   **What to observe:** The specific energy turns positive and the predicted curve no longer closes.

Physical conclusion

Check the sign of the energy first, then use the angular momentum to understand the shape.

Starter path · Step 6 of 6

## From a falling ball to an orbit

You have reached the end of this path. [Browse every Motion & dynamics Lab →](https://sandboxphysics.com/topics/mechanics/#directory-search)

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)