## Thomas' Calculus 13th Edition

$v_0=\sqrt {\dfrac{G M}{r_0}}$ and $\sqrt {\dfrac{G M}{r_0}} \lt v_0 \lt \sqrt {\dfrac{2G M}{r_0}}$ and $v_0=\sqrt {\dfrac{2 G M}{r_0}}$ and $v_0 \gt \sqrt {\dfrac{2 G M}{r_0}}$
The eccentricity can be written as: $e=\dfrac{r_0^2v_0^2}{G M}-1$ We know that the orbit will be a circle when $e=0$ So, $v_0=\sqrt {\dfrac{G M}{r_0}}$ We know that the orbit will be an ellipse when $0 \lt e \lt 1$ So, $\sqrt {\dfrac{G M}{r_0}} \lt v_0 \lt \sqrt {\dfrac{2G M}{r_0}}$ We know that the orbit will be a parabola when $e=1$ So, $v_0=\sqrt {\dfrac{2 G M}{r_0}}$ We know that the orbit will be a hyperbola when $e \gt 1$ So, $v_0 \gt \sqrt {\dfrac{2 G M}{r_0}}$