Answer
1. $v_0=\sqrt {\dfrac{G M}{r_0}}$
2. $\sqrt {\dfrac{G M}{r_0}} \lt v_0 \lt \sqrt {\dfrac{2G M}{r_0}}$
3. $v_0=\sqrt {\dfrac{2 G M}{r_0}} $
4. $v_0 \gt \sqrt {\dfrac{2 G M}{r_0}} $
Work Step by Step
The eccentricity can be expressed as: $e=\dfrac{r_0^2v_0^2}{G M}-1$
The orbit will be a circle when $e=0$
$\implies v_0=\sqrt {\dfrac{G M}{r_0}}$
The orbit will be an ellipse when $ 0 \lt e \lt 1$
$\implies \sqrt {\dfrac{G M}{r_0}} \lt v_0 \lt \sqrt {\dfrac{2G M}{r_0}}$
The orbit will be a parabola when $ e=1$
$\implies v_0=\sqrt {\dfrac{2 G M}{r_0}} $
The orbit will be a hyperbola when $ e \gt 1$
$\implies v_0 \gt \sqrt {\dfrac{2 G M}{r_0}} $
