How do I isolate for h in a question that is -GmM=( Ri +h)(Ri x m x Vi^2 - 2GMm) / 2Ri

Question

This is for a physics question where Ru I the distance inital rom the core of the earth and Ri+h is the distance from core of the earth plus the distance the object is  falling. It would be so helpful if someone could help Me. 

Answer 1

There is no R_u in the given equation.  It looks like the given equation is

 

-GmM = (R_i + h)(R_imv_i² - 2GMm) / (2R_i)

 

I will assume the equation is correct.  You want to find h.  Note that h appears only in the factor (R_i + h) on the right hand side.  Note you can divide by m on both sides to eliminate m.  Do that, and then multiply both sides by 2R_i and get

 

-2R_iGM = (R_i + h)(R_iv_i² - 2GM)

 

Divide both sides by (R_iv_i² - 2GM) and get

 

-2R_iGM / (R_iv_i2 - 2GM) = R_i + h

 

Finally,

 

h = [-2R_iGM / (R_iv_i² - 2GM)] - R_i