Lambert(Guidance) Equation Derive (Velocity) from equations (Need help-

[cemfrost]

1. The problem statement, all variables and given/known data

 

[Moach]

wolfram alpha?


i'm confuddled.... V doesn't even appear on two of the three equations above... plus, i'm not quite sure i understand the question :shrug:


could you elaborate on the problem and what you're trying to achieve?




EDIT :

how rude of me, i almost forgot -- Welcome to the Orbiter forums!

 

[RGClark]

1. The problem statement, all variables and given/known data

I take it you're asking to prove the last equation given the first four. First note equation 4 is equivalent to equation 2, so that's not adding new information. Equation 3 is also just the value of the constant gm so that is not needed to do the algebra either.
So it's just the first 2 equations that are needed to derive the value of V in the last equation. Notice that the λ does not appear in the last equation you are trying to derive but it does appear in the first two. That suggests using these two equations to eliminate the λ.
The first equation tells you explicitly what λ is so use that equation to substitute in for λ in the second equation. Now notice you have an equation with a V in it with all those other cosines and r's in it. So unwrap this equation to solve for V:

You have eq. 2 at the start:

Now when you use eq 1 to substitute in for λ, it becomes:

Now multiply both sides by cosγ and it becomes:

Now subtract the cos(ϕ+γ) and it becomes:

Now you see the V^2 in the denominator can be brought over to the left side up in the numerator, the entire left side can be brought over and put in the denominator on the right, and the gm term which is divided in the denominator can be brought up into the numerator while still on the right side. The equation then becomes:

Now do the square-root of both sides.

How do these equations arise?

Bob Clark

 

[cemfrost]

Thank you for everything, our solution is same

---------- Post added at 12:18 PM ---------- Previous post was at 01:27 AM ----------

My solution

---------- Post added at 12:18 PM ---------- Previous post was at 12:18 PM ----------

My solution was added ..