TL;DR:
An orbit is a fall with enough sideways speed that the ground curves away as fast as you drop.
Gravity is the centripetal force, so
mv²/r = GMm/r²andv = √(GM/r). Near Earth's surface that is about 7.9 km/s.The Moon falls toward Earth all the time, about 1.35 mm each second, and its sideways motion keeps it from arriving.
The Moon is pulled toward Earth by gravity every second of every day. So why does it not crash into us? The answer is that it is falling, and it has been falling for billions of years, but it is also moving sideways so fast that it keeps missing.
Throw a ball faster and faster
Picture a tall mountain and a ball thrown horizontally from the top. At a low speed it lands near the base. Throw harder and it lands farther out. In each case it falls with the same vertical acceleration, the one you met in why everything falls at the same rate, but covers more ground while it falls.
At some speed, the ball travels so far sideways that the ground curves away beneath it. It keeps falling, but the surface drops away by the same amount, so it never lands. That is an orbit.
Gravity as the centripetal force
An object moving in a circle needs a net force toward the centre, the centripetal force, of size mv²/r. For an orbit, gravity provides it: GMm/r² = mv²/r. Here G is the gravitational constant, M the mass of the planet, m the orbiting mass and r the distance from the planet's centre.
The small mass m cancels, and one factor of r too, leaving v = √(GM/r). Notice that the speed of a circular orbit depends only on the planet's mass and the distance. A bigger satellite goes at the same speed as a small one at the same height. Farther out means slower.
Worked example: how fast must the ball go?
Ignore air and mountains and consider an object just above the surface. Use Earth's radius R = 6.371 × 10⁶ m and g = 9.8 m/s². Near the surface, gravity gives acceleration g, so mg = mv²/R and v = √(gR).
Fall in one second from rest:
h = ½gt² = ½ × 9.8 × 1² = 4.9 m.How far must the ball travel sideways so that Earth's surface drops 4.9 m? For a circle, the drop over a horizontal distance d is about
d²/(2R). Setd²/(2R) = 4.9.Solve:
d² = 2 × 6.371 × 10⁶ × 4.9 = 6.243 × 10⁷, sod ≈ 7900 m.Covering 7900 m in one second means
v ≈ 7.9 km/s.Direct formula as a check:
v = √(gR) = √(9.8 × 6.371 × 10⁶) = √(6.244 × 10⁷) ≈ 7902 m/s.
Both methods agree at about 7.9 km/s, which is roughly 28,000 km/h (7.9 × 3.6 = 28.4 thousand). The time for one lap: T = 2πR/v = 2π × 6.371 × 10⁶ / 7902 ≈ 5070 s, about 84 minutes. Real low-Earth-orbit satellites fly higher, above the air, so they are a little slower and take a bit longer.
The Moon's tiny fall
Take the Moon's distance as 3.84 × 10⁸ m and Earth's GM = 3.986 × 10¹⁴ m³/s². Its orbital speed is v = √(GM/r) = √(3.986 × 10¹⁴ / 3.84 × 10⁸) = √(1.038 × 10⁶) ≈ 1020 m/s, about 1.0 km/s.
Its acceleration toward Earth is a = GM/r² = 3.986 × 10¹⁴ / (3.84 × 10⁸)² ≈ 0.0027 m/s². In one second it falls ½ × 0.0027 × 1² ≈ 0.00135 m, about 1.35 mm. At the same time it moves 1.02 km sideways, and over that stretch the curved path bends away from a straight line by exactly that 1.35 mm. The fall and the sideways motion balance.
The Moon is about 60 times farther from Earth's centre than we are. Gravity falls with the square of distance, so the pull should be 1/60² of 9.8, which is 9.8 / 3600 ≈ 0.0027 m/s². The same value. That match was an early triumph for Newton's law.
Why astronauts float
Astronauts on the space station are not beyond gravity. At that height gravity is still nearly as strong as on the ground. They float because they and the station fall together around Earth, so nothing pushes on their feet. Weightlessness is the feeling of free fall.
How to check your answer
Units.
√(m³/s² ÷ m) = m/s.Two routes. Get the speed from the fall geometry and from
√(gR)and compare.Trend. A larger radius must give a slower speed and a longer period.
Scale. The result should be thousands of metres per second near Earth, not hundreds or millions.
Mistakes that cost marks
Measuring r from the surface instead of the centre.
Thinking there is no gravity in orbit, or that the centripetal force is a second force on top of gravity.
Believing a heavier satellite needs more speed.
Mixing kilometres and metres in
v = √(GM/r).Thinking an orbiting object is pulled outward. It is pulled inward and moves sideways.
Where you will use it next
Orbits lead to Kepler's laws, escape speed and the reasoning behind satellite design. Keeping track of units, as in units and dimensional analysis, makes numbers like 10¹⁴ manageable.
Frequently asked questions
What happens if an orbiting object slows down?
It falls to a lower, faster orbit or drops into the atmosphere. Satellites in very low orbit slow down because of thin air and need boosts.
Are orbits always circles?
No. They are generally ellipses. A circle is the special case where the speed is exactly right at that distance.
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