I'm having trouble following that, partly because I can't tell whether letters are constants or units, and I don't know what the dt is doing.

Ahh, yes,

I always hate it when letters are poorly defined.

Actually, in the equations I wrote, the only real variable is t = time.

g is the gravitational constant, 9.8 m/s

^{2}, where m is meters, and s is seconds. km would be kilometers.

So, like pi, (π), g does not vary, although sometimes I'll round it to 10 m/s

^{2}. Your gravity might vary a little based on your location, for example on the moon. But, then it can be written as a fraction of Earth's gravity, g.

m = meters

s = seconds

km = kilometers

I am expressing Distance as a function of time Distance(t).

Likewise Velocity is a function of time Velocity(t).

And Acceleration is a function of time Acceleration(t).

d.... That signifies change. in something. You will also see delta (Δ), which means change in time, or the symbol (∂) for partial derivatives.

Why the dt? I suppose it tells you that you are evaluating a function with respect to the change of time. And, thus, it is required to be a part of integration.

Hmmm

Relating Distance, Velocity, and Acceleration.

if f(t) is a distance function with respect to time t.

Then Velocity(t) is a function of the rate of change of the distance at any time t.

I.E. If you are driving from LA to New York. If Velocity = 0, that means you are parked, and distance is not changing.

If the Velocity is > 0, then you are going forward.

If the Velocity is < 0, then you are going backward.

Anyway, the Velocity(t) is the rate of change of the distance function, or the derivative of the distance function d(f(x))

Acceleration tells you how fast you are changing the velocity function (how much you've pressed the gas pedal down). So, it would be the slope of the velocity function at any given time. Acceleration(t) = d(Velocity(t)) = d(d(f(t))

I suppose I forgot that when one takes derivatives, one looses one's constant factors. So, taking integrals, one should add them back in when doing integrals. I.E. What is the initial distance or velocity, which in some cases can be set to zero.

Anyway, the mathematics is relevant for scientific discussions. However, one can often understand a lot about a subject without needing anything very complex.