Question

Why do planets move faster around a sun with a larger mass?

Answer 1

F - for simplification - we assume the orbits are circular (which they approximately are), we have a centrifugal force of mv2Rmv2R, where m is the mass of the planet, v its speed, and R the distance from the (center of the) sun. This force must be equal to the centripetal force, which comes from the sun's attraction, and it is mGMR2mGMR2. Here, M is the mass of the sun, and G is a constant. We don't care about the actual numbers, so we write C be the product MG and observe that it is a constant independent of the orbit. These forces have to cancel out, so we have v2R=CR−2v2R=CR−2, or v2=CR−3v2=CR−3, so we have that the speed is proportional to R−32R−32. Which means that if you decrease the distance to the sun, the speed goes up.