Newton's Cannonball and the Cosmic Velocities — 3D Lab Simulation
Fire a cannon from a tall mountain and raise the speed to watch the ball fall, orbit, escape Earth and finally leave the Solar System
Theoretical Foundation & Historical Context
In the Principia (De mundi systemate, 1687) Newton imagined a cannon on a very tall mountain: the faster it fires, the farther the ball lands, until at some speed it never lands and circles the Earth. This simulation places the cannon at altitude h (default 100 km, above the atmosphere so there is no drag) and fires impulsively with speed v at flight-path angle γ. The predicted path is the exact two-body conic, while the ball is integrated with a fixed-step velocity-Verlet scheme. In the heliocentric view, once v exceeds v₂ we use patched conics: the leftover v∞ is added to Earth's 29.78 km/s orbital speed. The launch is impulsive; the cannonball or rocket shape is purely visual.
Law: Universal gravitation and the first, second and third cosmic velocities | Scientist: Isaac Newton | Year: 1687
Key Learning Objectives & Competencies
- ✓ Explain Newton's thought experiment and why a fast enough ball orbits instead of landing.
- ✓ Derive v₁ = √(GM/r), v₂ = √2·v₁ and v₃ from energy and circular-orbit arguments.
- ✓ Classify a trajectory as ellipse, parabola or hyperbola from its specific energy.
Mathematical Formulations & Differential Equations
v₁ = √(GM / r₀) ≈ 7,9 km/s (mặt đất)
Gravity supplies the centripetal force: v²/r = GM/r². At h = 100 km, v₁ ≈ 7.85 km/s
ε = v²/2 − GM/r = 0 ⇒ v₂ = √2 · v₁ ≈ 11,2 km/s
Specific energy equals zero: a parabolic path, reaching infinity with zero speed
v₃ = √(v₂² + v∞²), v∞ = (√2 − 1) · 29,78 km/s ≈ 12,3 km/s ⇒ v₃ ≈ 16,6 km/s
Launch along Earth's motion; the heliocentric speed must reach √2 · 29.78 ≈ 42.1 km/s
ε < 0: elip (e < 1) · ε = 0: parabol (e = 1) · ε > 0: hypebol (e > 1); e = √(1 + 2εh²/(GM)²)
For a horizontal launch, v < v₁ gives an ellipse that meets the ground so the ball lands; v₁ ≤ v < v₂ gives an ellipse around Earth
Physical Parameters & Units of Measurement
| Symbol | Physical Quantity | SI Unit | Default |
|---|---|---|---|
| h | Launch altitude above the surface (km) | — | — |
| v | Impulsive launch speed (km/s) | — | — |
| γ | Flight-path angle above the local horizontal (degrees) | — | — |
| ε | Specific orbital energy v²/2 − GM/r (MJ/kg) | — | — |
| e | Eccentricity of the conic | — | — |
| v∞ | Hyperbolic excess speed far from Earth, √(v² − v₂²) | — | — |
Real-World & Industrial Engineering Applications
Choosing the launch speed of low Earth orbit satellites (about 7.8 km/s).
Planning escape burns for lunar and interplanetary probes.
Estimating the energy needed for missions like Voyager that leave the Solar System.
3-Step Controlled Experimental Protocol
- Step 1: Baseline Setup — Initialize the simulation in standard conditions and observe undisturbed system dynamics.
- Step 2: Parameter Variation — Gradually modulate physical parameters using the control panel sliders.
- Step 3: Verification & Phase Space Analysis — Compare phase space trajectories and conservation metrics against analytical ODE solutions.
Frequently Asked Questions (FAQ)
Why does a ball fired horizontally at 7.9 km/s not hit the ground?
It is still in free fall, but the ground curves away at the same rate: each second the ball falls about 4.9 m while Earth's surface drops by the same amount, so the altitude stays constant.
Does the simulation include air drag?
No. The cannon sits at a default 100 km, above the atmosphere, and the launch is impulsive, so there is no drag or thrust phase. At real ground level drag would prevent reaching the first cosmic velocity.
Why is v₃ only about 16.6 km/s and not 42.1 km/s?
Because Earth already moves around the Sun at 29.78 km/s. Adding only v∞ ≈ 12.3 km/s along that direction reaches 42.1 km/s; combined with the cost of leaving Earth this gives v₃.
Related Simulations & Experiments
Interactive Simulation: Interactive real-time 60 FPS WebGL simulation is active in your browser. Rotate, zoom, and inspect physical apparatus components with full camera freedom.