Firstly, I know that the equation for the escape velocity is $$v_{\text{escape}}=\sqrt{\frac{2\,GM}{r}}\tag{1}$$ and understand it's derivation.
The following is such a simple derivation; for a test body of mass $m$ in orbit with a massive body (assumed to be spherical) with mass $M$ and separation $r$ between the two body's centres. Equating the centripetal force to the gravitational force yields;
$$\frac{mv^2}{r}=\frac{GMm}{r^2}\tag{2}$$
which on simplificaton, gives
$$v=\sqrt{\frac{GM}{r}}\tag{3}$$
What I would like to know is why eqn $(3)$ is not a valid escape velocity equation?
Or, put in another way, mathematically, the derivation in $(2)$ seems sound; yet it is out by a factor of $\sqrt{2}$. What is 'missing' from the derivation $(2)$?
EDIT:
As I mentioned in the comment below, just to be clear, I understand that equation $(3)$ will give the velocity required for a bound circular orbit. But to escape it should follow that the test mass has to move at any speed that is infinitesimally larger than $\sqrt{\frac{GM}{r}}$ such that $$v_{\text{escape from orbit}}\gt\sqrt{\frac{\,GM}{r}}$$
So in other words eqn $(3)$ gives the smallest possible speed for a bound circular orbit. I referred to this as the 'escape speed'; since speeds larger than this will lead to a non-circular orbit, and larger still will lead to an escape from the elliptical orbit.
So my final question is; do the formulas $(1)$ and $(3)$ actually give the highest possible speed not to escape orbit rather than the 'escape speed' itself?
Thank you to all those that contributed these answers.
Best Answer
First things first, in Newtonian mechanics, when an object travels around its host (e.g. a planet to a star), it follows an orbit that is a conic section: either an ellipse, parabola, or hyperbola. A circle is a special case of an ellipse. An ellipse is a bound orbit while the other two are unbound.
Your derivation assumes a circular orbit. However a perturbed circular orbit doesn't become hyperbolic - it becomes elliptical. In other words, if you take an object that's currently moving in a circle and get it to move a little faster, it doesn't shift to a hyperbolic orbit. It's still bound to the host.
The escape velocity is the minimum velocity needed for the object to become unbound. An object needs to move at $v \geq \sqrt{\frac{2GM}{r}}$ to be on a hyperbolic orbit.