[Coda]

It was probably using a Greek font with the letter D instead of using the Unicode uppercase delta symbol.

If it helps a light bulb turn on in your head, consider this: If you know how far the object moved in a certain span of time -- that is, if you know Δx and Δt -- then you can use the formula for velocity you already know: velocity = distance / time. This is actually measuring the average velocity. No matter what path the object took, whether it was moving at a constant speed or if it sped up or slowed down or even went two-steps-forward-one-step-back the whole time, as a whole, the velocity balances out to this value.

So OF COURSE if you go for that period of time averaging that velocity you'll go that distance!

So that just leaves the special case of constant acceleration. The assertion is that the average velocity under constant acceleration is the average of the initial and final velocity (that is, (vi + vf)/2). "But Coda," someone might ask, "what about all of the velocities in between? Why don't they count?" To which I say: Consider any two points in time equidistant from the middle. The initial and final velocities are obviously such a pair, but you might look at 25% into the time span and 75% into the time span, or 10% and 90%. No matter what pair you pick, because it's constant acceleration those two points will always average to the same value -- the velocity at exactly 50%.

As for why mass * velocity = change in momentum = force * time, well, dimensional analysis is a useful tool here. The units of velocity is m / s. The units of momentum is kg * m / s. The units of force is kg * m / (s^2). And obviously the units of time is s and the units of mass is kg. So...

mass * velocity = momentum = force * time
kg * m / s = kg * m / s = kg * m / s / s * s

(If this makes you wonder if these equations can be used to deal with a change in mass, then good on you for being curious -- the answer is "yes, that's what they use for rockets as they burn fuel.")

As for the names "momentum" and "impulse", that's just something you have to remember. "Momentum" is named that because it more or less matches up with the vernacular usage of the word. The term "impulse" was chosen because the common usage of the word means "push" or "drive".