Mechanics

Equal areas sweep

Shaded wedges from M to the trail sweep equal areas in equal times.

Equal areas sweep — interactive Mechanics simulation. Shaded wedges from M to the trail sweep equal areas in equal times. Free browser-based virtual physics lab with live SI measurements and a guided tutorial.

Kepler's second law

A line from the central mass to the satellite sweeps equal areas in equal times — the satellite moves faster when closer.

Angular momentum conservation explains why orbital speed increases near periapsis. Equal-area sweeps are visible as shaded wedges on the stage in this preset.

• dA/dt = constant
• L = mvr sin θ = constant

Investigation brief

Plan the question before you open the lab

The brief mirrors the prerendered page: driving question, competing predictions, variable roles, governing laws, setup, analysis and extension prompts remain visible and in this order.

Driving question

A satellite on an elliptical orbit is launched sideways at perihelion (closest approach). As it coasts out to aphelion and back, does the line joining it to the central mass sweep out area faster near perihelion, faster near aphelion, or at the same rate everywhere?

Predictions to weigh

• The radius line sweeps area fastest near perihelion, where the satellite moves quickest.
• The radius line sweeps equal areas in equal times — the same rate everywhere on the orbit.
• The radius line sweeps area fastest near aphelion, where the satellite has the most room to move.

Variable roles

What you set:

• Perihelion radius r_p (m)
• Aphelion radius r_a (m)

What you measure:

• Escape speed at perihelion v_esc (m/s)
• Specific angular momentum h (m²/s)

How the investigation runs

1. Open the equal-areas simulation and press Reset. The central mass sits at the middle of the stage with gravitational parameter GM = 40 m³/s², and the shaded sector wedges trace the area the satellite's radius line sweeps out.
2. Enable the specific-angular-momentum probe (h) and the eccentricity probe (e) on the satellite so both invariants are shown live.
3. For each trial below, set the perihelion radius r_p and aphelion radius r_a, launch the satellite from perihelion with a purely sideways (tangential) velocity, press Play, and let it complete at least one full orbit before recording the settled h reading.

Governing equation

Escape Speed — v_esc = √(2GM / r)

The minimum speed at radius r that would let the satellite break free of the central mass. The satellite's real perihelion speed stays below this value, so the orbit remains bound — a closed ellipse that sweeps equal areas in equal times.

Circular Orbital Speed — v = √(GM / r)

The tangential speed that produces a perfect circle at radius r. Launching faster than this at perihelion (as every elliptical trial here does) sends the satellite outward to a distant aphelion before gravity pulls it back.

What the printable worksheet asks students to work out

• For one orbit, note the h reading at perihelion and again at aphelion. Within reading error they should match. Using h = r·v, explain why the satellite must move slower when r is large (aphelion) and faster when r is small (perihelion) to keep h constant.
• The area swept per unit time is dA/dt = h/2. Since h is constant, equal areas are swept in equal times. Use your measured h for the first orbit to compute dA/dt, and confirm h = √(2·GM·r_p·r_a/(r_p + r_a)) reproduces the value you read off the sim.
• Compare the satellite's perihelion speed (v_p = h/r_p) to the escape speed v_esc you computed for each row. Is v_p above or below v_esc, and what does that tell you about whether the orbit stays bound?

Where this shows up beyond the lab

• Halley's Comet whips past the Sun in months but takes decades to crawl back from the outer solar system. Using the equal-areas law, explain why a comet spends so little of its orbit near the Sun and so much of it far away.
• Specific angular momentum h = r·v carries no mass term. If you doubled the satellite's mass but launched it identically, would the measured h change? Would its full angular momentum L = m·r·v change? Explain the difference.

• AP Physics 1 — Unit 6: Energy and Momentum of Rotating Systems
• AP Physics C: Mechanics — Unit 6: Energy and Momentum of Rotating Systems
• IB Physics — A.4 Rigid body mechanics
• IB Physics — D.1 Gravitational fields
• General High School Physics — Rotation, gravitation & orbits
• NGSS High School Physics — Gravitational and electrostatic forces

1. Welcome to Grav Equal Areas
2. Select the satellite
3. Press Play
4. Equal areas
5. Open the data
6. You did it!

Open the interactive simulation to build the scene, press Play, and explore with live measurements and a guided tutorial.

Open Interactive Lab →

Download lab sheet →

Kepler's Laws of Planetary Motion