[Physics] Why does velocity of electron increases with increase atomic number in the Bohr model of the hydrogen and hydrogen like atoms

Question

I already know mathematical proof which states velocity of electron increases with increase in atomic number,but what is the intuition behind it?

Answer 1

The same reason the Earth would move faster should the mass of the sun increase: centripetal force grows. In the case of Bohr's model, the force grows as $\sim Z$,

$$ F_e = \frac{k Ze^2}{r^2} $$

Newton's law thus results in

$$ \frac{mv^2}{r} = F_e = \frac{k Ze^2}{r^2} $$

leading to $v \sim Z^{1/2}$