The mathematics behind every number in this topic

Understand gave you weight = mass × g and treated g as a fixed number for each world. This layer asks the harder question: where does g itself come from?

The answer is Newton's full law of universal gravitation, with an actual number, G, that lets you calculate the force between any two masses anywhere — a pencil and a planet, two people on a bus, or the Sun and the Earth. From that one equation you can derive g on any world, the exact speed of any orbit, escape velocity, Kepler's laws, and the reason nobody has ever fully explained why gravity and inertia use the same mass.

Newton's law of universal gravitation, in full:

F = G × m₁ × m₂ ÷ r²

F is the force in newtons, m₁ and m₂ are the two masses in kilograms, r is the distance between their centres in metres, and G is the universal gravitational constant — the same number everywhere in the universe, for every pair of masses that has ever existed.

G = 0.0000000000667, or 6.674 × 10⁻¹¹ N·m²/kg².

G is almost unimaginably small. That smallness is why gravity between everyday objects is too weak to notice, and why it takes a mass as big as a planet before the force becomes obvious.

Understand gave you the orbital speed at Earth’s surface as a fact: about 7.91 km/s. Here is where that number actually comes from.

For an object moving in a circle, staying on the circle requires a centripetal force pointing towards the centre, equal to m × v² ÷ r. For an orbiting object, gravity is that force. Setting the two equal:

G × M × m ÷ r² = m × v² ÷ r

Notice m — the orbiting object's own mass — appears on both sides and cancels immediately. This is the same "mass cancels" idea from Understand, showing up a third time, in orbits. Rearranging what is left:

v = √(G × M ÷ r)

This is the exact formula behind every orbital speed in this topic.

Every orbit lab so far has shown circles alongside stretched ellipses without dwelling on one detail: on an ellipse, speed is not constant. A spacecraft on an elliptical path moves fastest at its closest point to Earth (perigee) and slowest at its farthest point (apogee).

This is Kepler's second law, found (like his third) purely from observation, decades before Newton: a line from the planet to the Sun sweeps out equal areas in equal times. Close to the Sun, that line is short, so the planet must move quickly to sweep the same area; far away, the line is long, so it can move slowly and still sweep the same area.

Investigate showed you the two numbers side by side. Here is the reasoning behind each one, precisely.

The predicted acceleration comes from taking the surface value g = 9.8 N/kg and scaling it down by the inverse square of how much further away the Moon is than the Earth’s surface: a distance ratio of 384,400 ÷ 6,371 ≈ 60.3.

predicted a = 9.8 ÷ 60.3² ≈ 9.8 ÷ 3,640 ≈ 0.00269 m/s²

The observed acceleration treats the Moon as a satellite on a circular path and uses pure kinematics: for circular motion, centripetal acceleration = v² ÷ r. The Moon’s orbital speed, from its known circumference and 27.32-day period, is about 1.02 km/s, and its distance is 384,400 km:

observed a = v² ÷ r = (1,023)² ÷ 384,400,000 ≈ 0.00272 m/s²

The two agree to within about 1.2 % — well inside the accuracy Newton could have expected from 17th-century measurements of the Earth’s size and the Moon’s distance.

Go back to Understand’s Chapter 4: mass cancels in free fall because gravitational force and inertia both scale with the same mass. Stated as an equation:

gravitational mass (in F = G M m ÷ r²) = inertial mass (in F = m a)

These come from two entirely different definitions. Gravitational mass measures how strongly a world pulls on an object. Inertial mass measures how stubbornly the object resists being pushed, by anything — a rocket engine, a cricket bat, a car crash — with no gravity involved at all. There is no obvious reason these should be the same number.

Every experiment ever performed says they are equal, to a precision better than one part in 10¹⁴ in the most careful modern tests using satellites built specifically to check it. Newton knew about the equality and used it, but had no explanation for why it should be true. It sat, unexplained, for over two centuries.

Newton’s law says g = G M ÷ r², which depends only on distance from the centre — so in principle g should be identical at every point on a perfectly spherical, non-spinning Earth. Real measurements show small but very real differences, for two reasons.

The Earth is not a perfect sphere. Spinning for billions of years has flattened it very slightly, so the equator bulges outward: about 21 km further from the centre than the poles. Since the equator is further from the centre, gravity there is very slightly weaker.

The Earth is spinning. Standing at the equator, you are travelling in a circle nearly 1,670 km/h fast, which requires a small centripetal force pointing inward — supplied by gravity, leaving slightly less "spare" force to press you onto a scale. At the poles, you are on the spin axis and feel none of this effect.

Both effects push the same way: weaker gravity at the equator, stronger at the poles — about 9.78 N/kg against 9.83 N/kg, a difference of roughly half a percent.

Investigate showed what ISRO’s missions did — raise an orbit gradually, then transfer, then insert. Here is a little more of why it works, using the tools from this layer.

Each perigee burn adds speed at a single point of the orbit. Extra speed at any point of an orbit increases the total energy of that orbit, and more orbital energy means the object can coast further out before gravity pulls it back — raising the apogee, exactly as predicted in Investigate. Because the burn happens at perigee, and an orbit’s perigee is fixed by where you were and how fast you were going at that exact point, later burns keep returning to nearly the same perigee to push the apogee out again.

Gravity pulling on a whole ocean, rather than a single point, produces an effect Understand did not need: a differential pull, stronger on the near side of the Earth than on the far side, because the near side is closer to the Moon.

Because gravity follows an inverse-square law, this difference can be estimated by comparing the pull at the near side (384,400 − 6,371 km from the Moon) with the pull at the far side (384,400 + 6,371 km away). The two are close enough that the difference is small — but not zero, and definitely not nothing.