Mail Time: Gravitational potential energy in orbits

One of the Fundamentals Quiz questions in the leadup to the 2023 AP exams asked about gravitational potential energy in orbits: 

118. Satellite A is in low-earth orbit; satellite B orbits much farther from earth's surface.  Which has greater gravitational potential energy: the satellite A-earth system, or the satellite B-earth system?

A reader asks,

Can you explain why satellite B is greater.  Physics is not my strong suit and I understand why using Ug = mgh.  But when using Ug = - GMm / r , wouldn’t the gravitational potential energy be greater for A?  I’m a little confused.

Ah, this is the million dollar question in gravitation.  The negative sign is extra important here.  Energy is a scalar - the negative sign doesn't just indicate direction, as it does for force or acceleration.  A potential energy of -50 J is LESS than a potential energy of -25 J!  

So, for the satellite farther away, the denominator of your equation is greater, the whole fraction has a smaller absolute value... but the negative sign matters.  The negative sign ensures mathematically that objects farther away from each other have larger gravitational potential energy .

For AP Physics 1, I just write this last sentence as a fact - you almost never need to use this equation for gravitational potential energy.  (This will be true for the 2024 exam; for the 2025 exams and beyond, students may well need to use the negative sign fluently.  We'll see. Either way, I'd *start* by using the fact and ignoring the equation.)  For AP Physics C, use of the negative sign in the gravitational potential energy equation is an important concept mathematically as well as conceptually.