# Gravitational Potential Difference

The gravitational acceleration vector $\overrightarrow{g}$ is analogous the electric field vector $\overrightarrow{E}$. By utilizing the definition of potential and potential energy derive an expression for gravitational potential difference between two points. What is the potential energy of a 5 kg mass raised to a height of 5 meters? What is the gravitational potential of this point relative to the ground?

Gravitational acceleration is

$g=(G*M)/R^2$     derived from $F=m*g=(G m M)/R^2$
Gravitational field is

$Γ(R)=GM/R^2 =g$

Gravitational potential at distance R is

$V(R)=-\int \Gamma d R=(GM)/R+Constant=g R+constant$

“Gravitational voltage” between points $R_1=R_{earth} + 5m$  and $R_2=R_{earth}+0 m$ is:

$U(R_1 R_2 )=g(R_1-R_2 )=+5g$

Potential energy (for mass $m=5 kg$) is

$W=m*U(R_1 R_2 )=5mg=5*5*9.8=245 J$

Observation: In reality since the field lines are toward the mass M (inversely that for electric field for positive charges), the gravitational potential is negative. Thus the potential energy of a mass above the earth becomes negative.