Mechanics
Elliptical orbit
Watch the path stretch into an ellipse — speed is fastest at periapsis (equal areas).
Elliptical orbit — interactive Mechanics simulation. Watch the path stretch into an ellipse — speed is fastest at periapsis (equal areas). Free browser-based virtual physics lab with live SI measurements and a guided tutorial.
Elliptical orbits
With the right speed, a satellite follows an ellipse instead of a circle. Speed is highest at periapsis and lowest at apoapsis.
Elliptical orbits have eccentricity e > 0. Energy is conserved (KE + PE), so the satellite speeds up near the central mass and slows at the far end of the ellipse.
- E = KE + PE = constant
- e > 0 for an ellipse
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 is launched sideways at radius r about a central mass, at a speed below the circular-orbit speed. Does it trace a closed ellipse, hold a perfect circle, or fly off and never return?
Predictions to weigh
- It settles into a closed, elongated ellipse and returns each orbit.
- It holds a perfect circle at the launch radius.
- It escapes the central mass and never comes back.
Variable roles
What you set:
- Gravitational parameter GM (m³/s²)
- Launch radius r (m)
- Tangential launch speed v (m/s)
What you measure:
- Escape speed v_esc (m/s)
- Specific orbital energy ε (J/kg)
- Eccentricity e
How the investigation runs
- Open the elliptical-orbit simulation and press Reset. The central mass sits at the middle of the stage with gravitational parameter GM = 40 m³/s².
- Enable the energy and eccentricity probes on the satellite so both invariants are shown live.
- For each trial below, place the satellite at radius r on the x-axis, give it a purely sideways (tangential) launch speed v, press Play, and let it complete at least one orbit before recording the settled probe readings.
Governing equation
Escape Speed — v_esc = √(2GM / r)
The minimum tangential speed at radius r that would let the satellite escape the central mass. A launch below this speed keeps ε negative and the orbit bound — an ellipse rather than an unbound trajectory.
Circular Orbital Speed — v = √(GM / r)
The tangential speed that produces a perfect circle at radius r. Launching slower than this (as in every trial here) drops the satellite inward after release, tracing an ellipse with the launch point as its far apoapsis.
What the printable worksheet asks students to work out
- Compute the specific orbital energy ε = v²/2 − GM/r for each row and confirm the sign. What does a negative ε tell you about whether the satellite is bound to the central mass?
- Compare each launch speed v to the escape speed v_esc you computed. The launch is tangential, so h = r·v; show that e = √(1 + 2εh²/GM²) reproduces the eccentricity you read off the sim, and explain why a launch nearer escape speed gives a more elongated ellipse.
Where this shows up beyond the lab
- For each launch radius, what tangential speed would turn the ellipse into a perfect circle? (Hint: v_circ = √(GM/r).) Is your table's launch speed above or below that value, and how does that explain the orbit's shape?
- Long-period comets have eccentricities close to 1. What does that imply about their speed at closest approach relative to the local escape speed — and why don't they quite escape the Sun?
- 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
- Welcome to Grav Elliptical Orbit
- Select the satellite
- Press Play
- Speed varies
- Open the data
- You did it!
Open the interactive simulation to build the scene, press Play, and explore with live measurements and a guided tutorial.
Kepler's Laws of Planetary Motion