Orbits and Gravity: Newton's Cannon and Kepler's Laws

One force for apples and the Moon

Newton asked a bold question: could the force that drops an apple also hold the Moon on its path? He proposed that any two masses attract each other with F = G·m₁·m₂ / r². Here m₁ and m₂ are the masses in kilograms, r is the distance between their centres in metres, and G = 6.674 × 10⁻¹¹ N·m²/kg² is the gravitational constant. Test it: the Moon is about 60 Earth radii away, so Earth's pull there should be 9.81 / 60.3² ≈ 0.00269 m/s². The Moon's actual motion gives 0.00272 m/s², a close match. The tale of an apple hitting Newton's head is a legend.

Newton's cannon

Newton imagined a cannon on a very high mountain, firing a ball sideways. With a slow shot the ball curves down and lands nearby. Faster, and it lands farther away. But the Earth is round, so the ground also curves away beneath the flying ball. Fire hard enough and the ground curves away as fast as the ball falls. The ball is still falling all the time; it just never gets any closer. In the animation, watch each shot land a little farther. In the sandbox, use the launch-speed slider and note where the shots end up.

The first cosmic speed

A circular orbit needs gravity to supply exactly the pull that bends the path round: v = √(G·M / r), where M is the mass of the Earth and G·M = 3.986 × 10¹⁴ m³/s². At ground level r = 6371 km, so v = √(3.986 × 10¹⁴ / 6.371 × 10⁶) ≈ 7.91 km/s, about 28 500 km/h. A real satellite flies above the air. The International Space Station, at about 400 km, moves at 7.67 km/s and circles the Earth every 92 minutes. Astronauts float not because gravity is gone (it is still about 8.7 m/s² there) but because they and the station fall together.

Ellipses and escape

Above 7.91 km/s the orbit stretches into an ellipse, an oval: the object climbs far from Earth, slows down and comes back. The faster the launch, the longer the oval. At 11.2 km/s, which is √2 times the first cosmic speed (7.91 × 1.414 ≈ 11.19), the path no longer closes. The object escapes the Earth for good, slowing to a halt only infinitely far away. This is the escape speed, also called the second cosmic speed. Below 7.91 km/s a shot from the ground is too slow and returns to the surface. Drag the sandbox slider through all three regions and read the orbit type.

Kepler's laws

Before Newton, Johannes Kepler studied the careful observations of Mars made by Tycho Brahe. Around 1609 he found that planets move on ellipses with the Sun at one focus, and that a line from the Sun to a planet sweeps equal areas in equal times. So a planet speeds up when it is close to the Sun: Earth moves at 30.3 km/s in early January and 29.3 km/s in early July. The ellipses of the planets are only slightly oval, but the rule is the same for a comet on a long, stretched path.

The third law

In 1619 Kepler added a third law: the square of the orbital period T grows as the cube of the average distance a. Measure T in years and a in Earth-Sun distances, and T² = a³. For Mars, a = 1.524, so T = 1.524^1.5 ≈ 1.88 years, matching the real 687 days. For Jupiter, a = 5.2, so T = 5.2^1.5 ≈ 11.9 years. Newton later showed why the law holds: it follows from the inverse-square pull of gravity, the same F = G·m₁·m₂ / r² that holds the Moon.

Gravity and orbits