Question

why is the curve radius minimum when a projected ballistic ball reaches the maximum height?

Answer 1

Let's draw the image, that describes body movement in the field of gravity

It's known that

$$a _ {c} = \frac {v ^ {2}}{R}$$

Where $a_{c}$ - is centripetal acceleration, $v$ - velocity, $R$ - curve radius.

So,

$$R = \frac {v ^ {2}}{a _ {c}}$$

It's easy to notice, that in the top of the trajectory $v$ is minimal (according to the energy conservation law), $a_{c}$ is maximal (and equal to $g$ ), so $R$ is minimum.