Newton's cannonball is a thought experiment that explains why satellites stay in orbit. Imagine a cannon on a very tall mountain firing cannonballs horizontally at different speeds. Slow shots fall back to Earth nearby, while faster shots travel farther before hitting the ground.
At just the right speed, the cannonball falls toward Earth at the same rate that Earth's surface curves away beneath it, so it keeps going around the planet.
The key idea is that orbit is not the absence of gravity, but continuous free fall under gravity. Gravity provides the centripetal force that bends the cannonball's path into a circle or ellipse. If the launch speed is too low, the path intersects Earth, and if it is high enough but below escape speed, the object becomes a satellite.
If the speed reaches escape velocity, the object can leave Earth's gravity well instead of returning.
Understanding Astronautics: Newton's Cannonball
The important change in an orbit is usually a change in direction, not a steady loss of speed. In empty space, an object keeps moving in a straight line unless a force acts on it. Gravity continually turns the moving object inward.
This turning requires acceleration even when the object's speed stays nearly constant. For a circular path, the required inward acceleration equals speed squared divided by the orbital radius.
A larger orbit has a larger radius, so it needs less turning each second. That is why satellites farther from Earth travel more slowly and take longer to complete one trip around the planet.
A real cannon could not place an object into orbit from Earth's surface. The lower atmosphere is dense enough to create enormous drag at orbital speeds. Air resistance heats the object, slows it down, and can destroy it.
Rockets solve this problem by rising through the thickest air before building up most of their sideways speed. They use several stages because carrying empty fuel tanks wastes energy. Even satellites in low orbit meet a thin trace of air.
Over months or years, this drag gradually lowers their orbit. Many small satellites eventually re-enter and burn up.
Most satellite paths are not perfect circles. A small change in launch speed or direction produces an ellipse. In an elliptical orbit, the satellite has a closest point called perigee and a farthest point called apogee.
It moves fastest near perigee because gravity is stronger there. It moves slowest near apogee. Energy shifts between motion and position during this journey.
When a satellite climbs away from Earth, it loses some speed. When it falls back inward, it gains speed. Engineers use this effect when planning transfers between orbits, since a well-timed rocket burn can change the farthest or closest part of a path.
Orbit depends on position, speed, and direction together. Speed alone does not tell the whole story. A spacecraft moving at the same speed can follow very different paths if it is at a different altitude or pointed in a different direction.
Launch sites near the equator gain a small helpful speed from Earth's rotation, so launches often head east when the mission allows it. The International Space Station, weather satellites, GPS satellites, and lunar probes all rely on these ideas. When studying orbits, pay close attention to units, the difference between mass and weight, and the meaning of free fall.
Free fall means gravity is the only major force acting. It does not mean gravity has stopped.
Key Facts
- Gravity pulls the cannonball downward while its horizontal velocity carries it forward.
- A circular orbit near Earth requires about v = 7.9 km/s if air resistance is ignored.
- Centripetal acceleration for circular motion is a = v^2/r.
- Gravitational acceleration from Earth is g = GM/r^2.
- For a circular orbit, GMm/r^2 = mv^2/r, so v = sqrt(GM/r).
- Escape velocity is vesc = sqrt(2GM/r), which is sqrt(2) times the circular orbit speed at the same radius.
Vocabulary
- Orbit
- An orbit is the curved path of an object moving around a planet, moon, star, or other body under gravity.
- Projectile
- A projectile is an object that moves through space after being launched, with gravity as the main force acting on it.
- Centripetal force
- Centripetal force is the inward force that keeps an object moving along a curved path.
- Escape velocity
- Escape velocity is the minimum speed needed for an object to move away from a body without falling back, ignoring air resistance and propulsion.
- Free fall
- Free fall is motion in which gravity is the only significant force acting on an object.
Common Mistakes to Avoid
- Thinking orbit means there is no gravity, which is wrong because gravity is what bends the path and keeps the object moving around Earth.
- Forgetting Earth's curvature, which is wrong because the cannonball can keep missing the ground only because the surface curves away beneath it.
- Confusing orbital speed with escape speed, which is wrong because orbital speed keeps an object bound while escape speed lets it leave without returning.
- Ignoring air resistance near Earth's surface, which is wrong because real cannonballs would slow down and burn up unless the thought experiment assumes no atmosphere or a very high launch point.
Practice Questions
- 1 A cannonball is launched horizontally from a very tall mountain at 2.0 km/s. If its horizontal speed stays constant for 10 s in an idealized no-air model, how far horizontally does it travel in that time?
- 2 Use v = sqrt(GM/r) to find the circular orbital speed at Earth's surface, using GM = 3.99 x 10^14 m^3/s^2 and r = 6.37 x 10^6 m. Give your answer in km/s.
- 3 Explain why increasing the cannonball's horizontal speed can change its path from a crash back to Earth into an orbit, even though gravity is still pulling it downward.