Orbital Mechanics: Initial velocity of a Parabolic trajectory If I want to calculate the initial velocity, say for a planet orbiting a massive body, can I use the vis-viva equation in terms of eccentricity, and plug $e = 1$, to obtain the initial velocity that would create a parabolic motion? 
$v = \sqrt{G \cdot M \cdot (\frac{2}{r}-\frac{1}{a})}$
and 
$a = \frac{\text{SemiMinor}}{\sqrt{1 - e^2}} $
.
Like wise, can I follow the same procedure with $e > 1$ for Hyperbolic trajectories? 
 A: The vis-viva equation holds for parabolas and hyperbolas, so yes, you can use it to obtain an initial velocity if you also know $r$, the distance to that initial point.
However, you would not be able to directly use the equation for the semimajor axis, $a$, that you provided. Plugging in $e = 1$ or $e > 1$ to that equation does not really give a result at all. The $\sqrt{1-e^2}$ in the denominator equals zero for $e = 1$ and is undefined for $e > 1$. So for a parabola, $a = ∞$ (making the $1/a$ term in the vis-viva equation equal to zero). For a hyperbola, $a$ is negative.
As a bit of a side note: If you're just trying to determine the initial velocity for an object starting at infinity, $v = 0$ for parabolic orbits (since the total energy of such an object must be zero; this can also be verified using the vis-viva equation, plugging in $a = ∞$, $r = ∞$). For hyperbolic orbits with an object starting from infinity, the value of $v$ of course depends on the orbital parameters, but any initial velocity greater than zero will result in a hyperbolic orbit.
