Orbits: Newton’s Cannon
Put a cannon on a mountain and fire sideways. Too slow and the ball falls back; fast enough and it falls around the planet forever; faster still and it escapes. Choose the speed and angle, then press play and watch the path.
What Is an Orbit?
Newton imagined a cannon on a very high mountain. Every cannonball falls towards the ground — but the ground curves away beneath it. Fire fast enough and the ball keeps falling without ever getting closer: it is in orbit. An astronaut on the space station is weightless for exactly this reason: they are falling all the time.
Below the circular speed the ball lands. Between the circular and the escape speed it follows an ellipse, coming back to the cannon once per lap. At the escape speed the path opens into a parabola and the ball never returns.
Trajectory
to scaleOrbit Formulas
μ = GM| Quantity | Formula |
|---|
Worked Techniques
Negative energy: a closed ellipse with semi-major axis a. Zero: a parabola. Positive: a hyperbola that leaves for good. The launch direction only changes the shape, not the energy.
The time for one lap depends only on the size of the ellipse. Low Earth orbits take about 90 minutes; the Moon, at 384 000 km, takes 27 days.
If the lowest point rp lies inside the planet, the cannonball hits the ground before completing the ellipse.
Sources & Verification
- H. D. Curtis, Orbital Mechanics for Engineering Students, 3rd ed., Butterworth-Heinemann, 2014 — Ch. 2 (The Two-Body Problem: the energy law; circular, elliptical and hyperbolic orbits) and Ch. 3 (Orbital Position as a Function of Time).
- D. Halliday, R. Resnick & J. Walker, Fundamentals of Physics, 10th ed., Wiley, 2014 — Ch. 13 (Gravitation).