How gravity works: weight, falling and orbits

In Discover you met gravity as a story: everything falls, the Moon keeps missing, astronauts float because they are falling. This layer turns that story into numbers and rules you can use.

By the end of it you will be able to work out how long a stone takes to reach the ground, what you would weigh on Mars, why a skydiver stops speeding up, how fast a satellite must travel to stay up, and why "zero gravity" is the wrong name for what astronauts feel.

Three ideas do almost all the work:

1. weight = mass × g — the pull of a world on a lump of matter.
2. Falling speeds up by g every second, so distance = ½ × g × t².
3. Gravity pulls harder between bigger masses and weaker across bigger distances.

Everything else is a consequence.

Force is a push or a pull. Its unit is the newton (N).

A newton is defined by what it does to matter: one newton is the force that makes a mass of one kilogram speed up by one metre per second, every second. Written compactly, 1 N = 1 kg·m/s².

Mass is the amount of matter in an object, in kilograms. Mass does two jobs at once, and it is worth naming both:

• Gravitational mass: how strongly gravity pulls on the object.
• Inertial mass: how stubborn the object is — how hard it is to get moving or to stop.

A loaded goods wagon is hard to push on Earth. It would be exactly as hard to push on the Moon, even though it weighs six times less there. Stubbornness does not care about gravity. Keep this in your pocket; it explains the biggest puzzle in Chapter 4.

Weight is the gravitational force on a mass, in newtons. It is not a property of the object; it is a property of the object and where it is standing.

The number 9.8 turns up with two different units, and both are correct.

9.8 newtons per kilogram (N/kg). Read this way, g tells you the pull: every kilogram of an object is tugged with 9.8 newtons. It is the version you want when you are working out weights.

9.8 metres per second per second (m/s²). Read this way, g tells you the acceleration: a freely falling object gains 9.8 metres per second of speed in every second of falling. It is the version you want when you are working out falls.

These are the same number for a deep reason, which is Chapter 4's business. For now, notice what the second reading actually says. "Metres per second, per second" is a rate of change of a speed. After one second of falling the stone is doing 9.8 m/s; after two, 19.6 m/s; after three, 29.4 m/s. The speed is not the thing gravity gives you — the increase in speed is.

Two short rules describe any fall that starts from rest, as long as air resistance can be ignored.

Speed after t seconds: v = g × t

Distance fallen in t seconds: d = ½ × g × t²

The first is just the meaning of g: add 9.8 m/s of speed for each second.

The second needs a moment's thought. Why the ½, and why t squared? Because the object is not travelling at its final speed the whole way — it started at zero and finished at g × t. Its average speed over the fall is halfway between, which is ½ × g × t. Multiply an average speed by the time and you get the distance:

distance = average speed × time = (½ × g × t) × t = ½ × g × t²

That is the whole derivation, and it is worth being able to rebuild it rather than memorising it.

Here is the puzzle stated properly.

A 10 kg rock is pulled by the Earth with ten times the force of a 1 kg rock: 98 N against 9.8 N. Ten times the pull. So surely it must fall faster?

No — and the reason is the second job mass does.

The 10 kg rock also has ten times the inertia. It is ten times harder to get moving. So it receives ten times the push and needs ten times as much push to produce the same result.

The two tens cancel exactly. Written out:

acceleration = force ÷ mass = (m × g) ÷ m = g

The mass appears on the top and on the bottom of the fraction and vanishes. Whatever number you put in, the answer is g. A grain of sand, a cricket ball, an elephant and a lorry all accelerate downwards at exactly 9.8 m/s².

This is the deep reason a hammer and a feather land together on the Moon. It is not a coincidence, and it is not about the air. It is because the quantity that makes gravity pull on you is the very same quantity that makes you hard to move.

Real falls happen in air, and air is not nothing.

Air resistance (drag) is the backwards push of air on anything moving through it. Three things control how big it is:

• Speed. Drag grows roughly with the square of the speed at everyday sizes: go twice as fast and the air pushes back about four times as hard. This is the key to the whole chapter.
• Frontal area. How much air you have to shove aside. A flat sheet of paper has a huge frontal area for its mass; the same paper crumpled has a tiny one.
• Shape. A smooth, tapering shape lets air close in behind it; a flat, blunt one leaves a churning wake that drags it back.

Now watch what those three do to a fall. At the start, speed is zero, so drag is zero, and the object accelerates at the full 9.8 m/s². As it speeds up, drag grows rapidly. The net downward force (weight minus drag) shrinks, so the acceleration shrinks. Eventually drag grows all the way up to equal the weight. Net force zero. Acceleration zero. The speed stops changing.

This finally explains, properly, the thing that fooled everybody for two thousand years.

Two objects of the same shape and size but different mass — say a hollow plastic ball and a solid steel ball of identical diameter — meet exactly the same drag at any given speed. But the steel ball has far more weight for the drag to balance, so it has to go much faster before the two match. Its terminal velocity is much higher.

So in air, the heavier of two identically shaped objects really does fall faster — not because gravity accelerates it more, but because it takes more drag to stop it accelerating.

Aristotle was not blind. He was generalising from a world full of air, and he never took the air away. Galileo's leap was to ask what would happen without the air, and to design experiments that got the air out of the way.

Isaac Newton was born in 1642 in Woolsthorpe, England. When plague closed his university in 1665–66, he went home and spent two years thinking — later calling them his years of greatest invention.

The apple story is real, but not as usually told. What almost certainly did not happen: an apple landing on Newton's head, and him instantly shouting "gravity!"

What Newton himself described, in a 1752 account by his friend William Stukeley: an apple fell in the Woolsthorpe orchard, setting him wondering why it always fell straight down — and then, the actual leap, how far up that pull goes. To the treetop? The clouds? All the way to the Moon?

That question, not the apple, is the discovery: nobody before had supposed the force dropping fruit and the force steering the heavens might be the same force.

Newton then calculated: if the pull weakens with distance in a particular way, he could predict how fast the Moon should be falling, and compare it with how fast it is falling. It matched. Twenty years later he published it all in the Principia (1687).

Newton's law of universal gravitation, in words:

Every object in the universe attracts every other object, with a force that grows in proportion to each of their masses and shrinks in proportion to the square of the distance between their centres.

Three things are packed in. Every object — not planets only, every pair of masses, always. Grows with each mass — double one mass and the force doubles; double both and it quadruples, why Earth's pull dominates your life but the building next door's does not. Shrinks with distance squared — double the distance and the force is not halved but one quarter; triple it and it is one ninth.

Measured from the Earth's centre, not its surface: at 6,371 km, g is 9.8; twice as far out, g drops to 9.8 ÷ 4 = 2.45. The formula and the symbol G wait in Go deeper.

Newton's cannon, done properly.

Put a cannon on a mountain so tall it pokes above the air, and fire horizontally. The ball leaves at some sideways speed and immediately falls at 9.8 m/s², exactly like a dropped stone. Two motions happen at once and do not interfere: steady sideways speed, and ever-faster falling.

The result is a curve — fire faster and it is longer and flatter, but the ball falls at the same rate throughout.

The Earth is round, so its surface curves away from any straight line at a fixed rate: go 8 kilometres and the ground has dropped roughly 5 metres below where you started.

But 5 metres is exactly how far you fall in one second! So a ball travelling 8 km sideways per second sees the ground fall away just as fast as it does — the gap never closes, and it falls forever without landing.

That speed — about 7.9 km/s, or 28,400 km/h — is the orbital speed just above Earth's surface. Slower, you land. Faster, you swing into a stretched oval. At about 11.2 km/s the ball never comes back: escape velocity.

A satellite is simply something put into orbit on purpose. India has done this since Aryabhata, its first satellite, in 1975; ISRO's PSLV and LVM3 rockets now launch payloads from Sriharikota for India and customers worldwide.

Where you put a satellite depends on the job, and two heights matter most.

Low Earth orbit, a few hundred km up. The ISS sits at about 400 km, needing about 7.7 km/s (≈27,600 km/h) for a circular orbit, with one lap taking about 92 minutes — a sunrise roughly every 90 minutes, about 16 times a day. Earth-observation and weather satellites live here too, close enough to see detail.

Geostationary orbit, 35,786 km up. Further out, orbits get slower. At exactly 35,786 km above the equator, one lap takes 23 hours 56 minutes — exactly Earth's spin — so the satellite keeps pace with the ground and appears to hang motionless.

That is why a dish antenna is bolted in place, aimed once at a satellite that never wanders.

This is the chapter to get right, because almost every popular account gets it wrong.

Astronauts aboard the ISS are not beyond gravity. At 400 km, gravity is about 8.7 N/kg, roughly 89 % of ground strength. A 70 kg astronaut is pulled with about 608 N up there, against 686 N at home. The Earth has barely let go.

So why do they float? Because weight is not what you feel — what you feel is the floor pushing back. Gravity pulls you down, the chair pushes up, and it is the chair's push you experience as heaviness. Remove the chair and you feel nothing while falling, the same odd stomach-drop as a lift starting down.

On the ISS, the station and everything inside it — astronaut, floor, water, pen — are all in free fall together. Nothing presses on anything. It is that lift-starting-down moment, stretched out for months.

The correct name is free fall; the right word for the environment is microgravity — tiny residual effects, not an absence of gravity.

Months of free fall have real effects on a human body, because our bodies quietly depend on being pulled.

• Bone. Weight-bearing bones — hip, spine, thigh — lose roughly 1 % of their mineral per month, since bone rebuilds itself in response to load, and there is none. Close to 6 % over six months.
• Muscle. Postural leg and back muscles, unemployed without gravity to hold you upright, shrink.
• Fluid shift. Gravity normally keeps fluids low in the body; in free fall they redistribute upwards, giving puffy faces and thin legs for the first days.
• Height. Spinal discs expand without the constant squeeze, and astronauts gain a few centimetres — lost again within days of landing.

The countermeasure is exercise: crews spend about two hours a day on a treadmill or resistance machine, re-creating gravity's loads. It helps a great deal, but does not fully replace a planet.