Question about the gravitational Potential Energy Law Proof

[Alwahsh]

In the Proof of the Gravitational Potential Energy Law , it says :

Consider two point masses , body . mass "m" and Earth mass "M" , where M remains fixed. The work done to move the mass "m" from point 1 to point 2 is given by :

W(1-2) = ∫FG . dr (All in vectors) = ∫FG Cos 180° dr ( All in Magnitudes ) where ...
and the proof just goes on until we reach Ur = - G (Mm/r)


Now my question is , why do we consider the direction of motion of the point mass opposite to the direction of the gravitational force and at the end of the law we say r1 ( which is the initial position ) is ∞ and r2 ( which is the final position ) is = to the distance between the point mass and the Earth , This means the direction of motion of the point mass should be in the same direction of the Earth's gravitational force not opposite to it and so we shouldn't add this Cos 180 and the law at the end is : Ur = - G (Mm/r) .



Thanks in advance
Alwahsh

 

[Nate810]

The direction of motion of the point mass is actually not opposite to the direction of the gravitational force. The integral of the gravitational force is taken along the line connecting the two masses. The integral is taken with respect to the distance between the two masses, and this distance is the same regardless of the direction of motion. The Cos 180 term is just there to take into account the fact that the force is negative and thus the work done is negative. So the final equation holds regardless of the direction of motion.