How to play Planet Scale
- Choose a world and read its mass M and radius R, or the astronaut's mass, on the card.
- Use g = G·M ÷ R² for the surface gravity, W = m·g for the weight, and t = √(2h ÷ g) for the fall time.
- Type your answer and press Drop. The ball falls 2 m while the scale weighs the astronaut.
- Compare the gravity chart with Earth's bar, and remember: weight changes from world to world, mass does not.
Why is gravity different on each world?
Newton's law of universal gravitation says a world of mass M pulls an object of mass m on its surface with F = G·M·m ÷ R², where R is the world's radius. Dividing by m gives the acceleration due to gravity at the surface, g = G·M ÷ R². A world with more mass pulls harder, and a bigger radius puts you farther from its centre, which weakens the pull with the square of the distance. That is why g is 9.82 m/s² on Earth, 3.73 m/s² on Mars, and only 1.62 m/s² on the Moon. Your weight, the force a scale measures, is W = m·g, so it changes from world to world. Your mass, the amount of matter in you, stays the same everywhere.
A focused learning model
Each world is a uniform sphere, so surface gravity is G·M ÷ R². There is no air resistance, and the planet's rotation is ignored. The super-Earth is an imagined planet with 8 times Earth's mass and twice its radius. The ball's touchdown is found exactly. Every drop uses the same 2 m tower and plays at a quarter of real speed.