Why do we need the hyperbolic excess velocity?

[TimeRip496]

$$V_{HE}=\sqrt{\frac{\mu}{a}}$$

What is the rationale for this formula when we can determine the change in velocity from Earth's orbit to transfer orbit using the vis-viva equation? Likewise, what is the use of defining the radius for the sphere of influence for interplanetary transfer?
$$R_{SOI}=R(\frac{\mu_{planet}}{\mu_{sun}})^{2/5}$$

I just don't understand the rationale with coming up with these terms when the delta v for the transfer can be determined by finding the velocity from initial, transfer and final orbit.

 

[trurle]

$$V_{HE}=\sqrt{\frac{\mu}{a}}$$

What is the rationale for this formula when we can determine the change in velocity from Earth's orbit to transfer orbit using the vis-viva equation? Likewise, what is the use of defining the radius for the sphere of influence for interplanetary transfer?
$$R_{SOI}=R(\frac{\mu_{planet}}{\mu_{sun}})^{2/5}$$

I just don't understand the rationale with coming up with these terms when the delta v for the transfer can be determined by finding the velocity from initial, transfer and final orbit.

Calculations involving sphere of influence are useful for patched-conic models - these are easiest way to analyze multiple-bodies trajectories.