PhysSandbox

Orbits and Kepler’s Laws in a Gravity Sandbox

A 45-minute guided lesson: students watch a miniature solar system to discover that inner planets move faster (Kepler’s third law), break a circular orbit into an ellipse to see the speed change along it (second law), explore a binary star orbiting its centre of mass, and explain a gravitational slingshot.

Level: Grades 9–12 · AP Physics 1 · IB Physics SL45 minSimulator used in this lesson: Gravity Sandbox

Learning objectives

  • Explain an orbit as continuous free fall: gravity supplies the centripetal force, GMm/r^2 = m v^2 / r.
  • State Kepler’s three laws and recognise each of them in the simulation.
  • Predict qualitatively what happens to an orbit when the speed or the attraction changes.
  • Use Kepler’s third law quantitatively: T^2 propto r^3 for bodies orbiting the same central mass.

Students should already know

  • Newton’s law of gravitation F = G M m / r^2.
  • Centripetal acceleration a_c = v^2 / r (from the circular-motion lesson or textbook).

Standards

  • NGSS HS-PS2-4
  • NGSS HS-ESS1-4
  • AP Physics 1 · Unit 3 (Circular Motion and Gravitation)
  • IB Physics · Topic D.1
Warm-up

1.Why doesn’t the Moon fall down?

~5 min

Newton asked what would happen to a cannonball fired horizontally from a very high mountain, faster and faster. Before opening the simulator, decide:

In the sandbox every body attracts every other body with an inverse-square force. Bodies leave a short trail of their last few positions — the same number of points for every body — so a longer trail means a faster body. You can pause with the Space bar, add a body at rest by clicking empty space, and drag bodies to move them. Units are arbitrary: compare, don’t convert.

Explore 1

2.A miniature solar system

~10 min

Three light planets circle a heavy star at three different distances. Each starts with exactly the speed needed for a circular orbit.

Your task

  1. Compare the three trails. Which planet moves fastest — the inner, the middle or the outer one?
  2. Pick a moment when the inner and outer planets are lined up on the same side of the star. Count how many orbits the inner planet completes while the outer one completes one. (Use the time-scale slider to slow things down if you need to.)
  3. Estimate the ratio of the two orbital radii from the screen. Is the ratio of periods bigger or smaller than the ratio of radii?
  4. Does the star stay perfectly still? Watch the small cross marking the centre of mass.

The inner planet is the fastest and the outer the slowest, yet the outer one also has the longest path — so its period is much longer. The outer radius is about 2.2 times the inner one, and the inner planet laps the outer one roughly 3 times per outer orbit: the period grows faster than the radius. That is Kepler’s third law: T^2 propto r^3, so T propto r^3/2 and 2.2^3/2 approx 3.3. (The sandbox softens gravity at very short range, so your count may come out a little lower.)

Gravity as centripetal force

( dfracG M mr^2 = dfracm v^2r \;Rightarrow\; v = sqrtdfracG Mr\)

Period of a circular orbit

( T = dfrac2π rv = 2π sqrtdfracr^3G M\)

Kepler’s third law

( dfracT^2r^3 = dfrac4π^2G M = textconst for one central mass\)

Explore 2

3.Breaking the circle: ellipses and Kepler’s second law

~10 min

Two equal stars orbit each other in circles. The G slider sets the strength of gravity; the stars were launched with the speed that fits G = 120. Changing G after launch is equivalent to suddenly changing the stars’ speed relative to what the orbit needs.

Your task

  1. Watch a few circular orbits first. Where is the centre of mass, and does either star ever cross it?
  2. Lower G to about 70. The orbits become ovals. Watch the trail of one star over a full loop: where on the ellipse is the trail longest (fastest), and where shortest?
  3. Now raise G to about 250. What happens? Press Clear, then Load binary pair to start again if the stars merge.
  4. Lower G all the way to 10 — do the stars still orbit, or do they escape?

With too little attraction for their speed, the stars swing outward and come back on an ellipse (Kepler’s first law: orbits are ellipses, with the centre of mass at a focus). Along the ellipse each star is fastest at the closest point and slowest at the farthest — the content of Kepler’s second law, equal areas in equal times. Too much attraction and the stars spiral inward and merge; too little and they fly apart on open paths and never return.

Kepler’s first law

Each orbit is an ellipse with the centre of mass at one focus.

Kepler’s second law

The line to the orbiting body sweeps equal areas in equal times ( Rightarrow v_textnear r_textnear = v_textfar r_textfar\)

Escape condition

( tfrac12 m v^2 ge dfracG M mr \;Rightarrow\; v ge sqrtdfrac2 G Mr = sqrt2\, v_textcirc\)

Explore 3

4.Falling in, flying by, slinging out

~10 min

An empty sandbox. Clicking adds a body at rest; the mass of new bodies is set by the slider.

Your task

  1. Set New body mass to its maximum and click once in the middle to make a “star”. Then set the mass to 20 and click somewhere to the side. What path does the small body follow? Why does it not orbit?
  2. Add two small bodies at rest at the same distance on opposite sides of the star. Do they reach the star at the same time? What if one of them is placed twice as far away?
  3. Clear, then place two big bodies at rest, a hand’s width apart. Describe their motion. Where do they meet?

A body released from rest has no sideways speed, so it falls straight into the star — orbiting needs tangential velocity, not just gravity. Two equal bodies released from rest fall towards each other and meet at their centre of mass, exactly midway. Gravity is mutual: Newton’s third law says the star is pulled towards the planet with the same force the planet feels (it just accelerates less because it is heavier).

Finally, a pre-built fly-by: a small probe sweeps past a heavy planet (a third body is also in the scene).

Your task

  1. Watch the small body’s trail as it approaches, swings round and leaves. Where is it fastest?
  2. Compare its trail far away *before* the encounter with its trail far away *after*. Is the speed the same, larger or smaller? What changed — the speed or the direction?
Worked example

5.Kepler’s third law with real numbers

~5 min

A geostationary satellite

A geostationary satellite orbits the Earth once per sidereal day, T = 86 164 s, staying above the same point on the equator. Find the radius of its orbit and its altitude above the surface. (G = 6.67 × 10⁻¹¹ N·m²/kg², M_E = 5.97 × 10²⁴ kg, R_E = 6 371 km.)

  1. Gravity provides the centripetal force: dfracG M mr^2 = m dfrac4π^2 rT^2, so r^3 = dfracG M T^24π^2.
  2. Numerically: G M = 6.67 × 10^-11 · 5.97 × 10^24 = 3.98 × 10^14\ textm^3/texts^2; T^2 = 7.42 × 10^9\ texts^2.
  3. r^3 = dfrac3.98 × 10^14 · 7.42 × 10^939.5 = 7.48 × 10^22\ textm^3 Rightarrow r = 4.22 × 10^7\ textm = 42\,200\ textkm.
  4. Altitude: h = r - R_E = 42\,200 - 6\,371 approx 35\,800\ textkm — about 5.6 Earth radii above the ground. Speed: v = 2π r / T = 3.07\ textkm/s.

r ≈ 42 200 km, altitude ≈ 35 800 km, v ≈ 3.1 km/s. Every geostationary satellite — TV, weather, GPS augmentation — sits in this one ring.

Quiz

6.Check your understanding

Answer each question, then press Check. Numeric answers from the simulator are accepted within the stated tolerance.

1.A planet in a circular orbit is in…

2.Mars orbits the Sun at 1.52 times the Earth’s distance. How long is the Martian year, in Earth years?

years±3 % accepted

3.Satellite A orbits at radius r with period 2.0 h. Satellite B orbits the same planet at radius 4r. What is B’s period, in hours?

h±3 % accepted

4.A comet on a long elliptical orbit is moving fastest when it is…

5.A satellite in a circular orbit fires its engine briefly to increase its speed (without changing direction). Which statements are true? (Select all that apply.)

6.Two stars of equal mass orbit each other. The centre of mass is…

7.Homework

  1. Three-body “dance”: three equal stars start on a symmetric orbit. Watch for a while. Is the motion stable forever? Then drag one star a little off its path and compare. Write two sentences about why the three-body problem is famous. Open simulator with these settings ↗
  2. In the mini solar system, slow the time scale to 0.3 and measure the period of the inner and outer planets with a stopwatch (time three laps each and divide). Compute T_outer / T_inner and compare with (r_outer / r_inner)^1.5 ≈ 3.3. Suggest a reason for any difference. Open simulator with these settings ↗
  3. Use Kepler’s third law to find the orbital period of the International Space Station (altitude 420 km, Earth radius 6 371 km) given that a geostationary satellite at r = 42 200 km has T = 24 h. Check against the real value of about 93 minutes.

Notes for teachers

  • Pacing: Warm-up 5 min, Explore 1 10 min, Explore 2 10 min, Explore 3 10 min, worked example 5 min, quiz 5 min. Explore 3’s slingshot part is the natural cut if time is short; it also makes a good opener for a later lesson on energy.
  • The sandbox uses arbitrary units and a softened force law at short range (the “Softening ε” slider), so period ratios come out a few percent below the pure Kepler value. The text tells students to expect “roughly 3” rather than 3.3; homework 2 asks them to explain the difference, which is a good place to mention the softening.
  • Trails hold a fixed number of recent positions (the Trail memory slider), so trail length is a direct speed indicator. This is the main observational tool of the lesson — point it out explicitly before Explore 1.
  • In Explore 2, the G slider stands in for “changing the speed relative to what the orbit needs”, because speeds cannot be edited directly. Students who object that G is a constant in nature are right — turn it into a discussion: lowering G at fixed speed is equivalent to raising speed at fixed G.
  • Misconceptions targeted: (1) “centrifugal force balances gravity in orbit” — quiz 1; (2) “heavier planets orbit faster/slower” — in Explore 1 the planet masses differ and it makes no difference to the orbit; (3) “a satellite needs to keep firing engines to stay up” — Explore 3 shows the opposite: it needs speed, not thrust.
  • On phones the canvas is small; the solar and binary scenes scale with it, but encourage landscape orientation for Explore 1.