What is the gravity/inertia question?

Question

Is the discussion about how gravity and inertia are the same, about the idea that when you hit a ball, it doesn't come back but when you throw a ball up, it does come back?

Answer 1

This discussion has nothing to do with a ball being hit or thrown. It is about why the inertial mass is equal to the gravitational mass (see also inertial vs. gravitational mass). But for explaining this I need to reach out more.

Inertial mass is how much a body refuses to be accelerated. To accelerate a body with acceleration $\mathbf{a}$ you need to apply a force $$\mathbf{F}_a=m_i \mathbf{a}.$$ The factor $m_i$ appearing here is called the inertial mass of the body.

On the other hand, gravitational mass is how much a body reacts to gravity. When a body is in a gravitational field of strength $\mathbf{g}$ then it is subject to the gravitational force $$\mathbf{F}_g=m_g\mathbf{g}.$$ The factor $m_g$ appearing here is called the gravitational mass of the body (or more precisely: the passive gravitational mass)).

Now it is known (already by Newton in 1680, extending on Galilei's concepts from 1600) that for all bodies these two masses ($m_i$ and $m_g$) are equal. And therefore we usually call it just the mass of the body. But why are these two equal? This is a simple but highly non-trivial question. And even Newton didn't have an answer for this.

It took until 1915 (Einstein's theory of general relativity) to get an explanation for this coincidence. Actually this theory begins with the equivalence principle, claiming that a gravitational field and and an acceleration are not distinguishable from each other. Or saying in other words: The gravitational force is also kind of an inertial force, and therefore proportional to the inertial mass.