Here are two common and useful ways of thinking about the quantity g:
g is the acceleration of objects near the surface of the earth: 10 meters per second gained (or lost) in 1 second
g is also the gravitational field strength near earth’s surface: 10 Newton’s of force pulling for each 1 kg
And here is a another, less common, but still useful way of thinking about it:
g is the “specific work” of gravity near the earth’s surface: 10 Joules of energy transferred for each 1 kg that moves 1 meter
By “specific work”, I mean something much like specific heat. Of course, it tells us about work and not heat, so it’s about changes in meters and not changes in temperature. And specific heat is characteristic of substances (e.g., copper or water), while specific work (at least in this context) is characteristic of planets. Different planets have different specific works (near to their surfaces) just as different materials have different specific heats. And just like the specific heats of materials aren’t really constant over a really wide range of temperatures, specific work of a planet is not really constant over whole range of distances. But it’s constant enough over a range to make the construct useful.
I realize I’m not the first person to have seen the parallel between specific heat and g (as a kind of specific work), but it’s a cool idea. And I should mention that this idea, which I had in my sleep, is the result of my brain mulling over Leslie’s class’s realization that falling objects gain 20 (m/s)^2 in every 1 meter.
Post script: I hadn’t really thought about the structural similarity of the equations this morning, but this provides another entry point for thinking about why it makes sense to think of g as the “specific work capacity”
Q = m c ΔT
W = – m g Δy
Of course this means that the force, “mg” is the (non-specific) work capacity of any particular object being lifted away from the earth. The word “capacity” also resonates with me thinking of the gravitational field as being capable of storing energy.