the FUN-damental Equation?

In class you used a formulaic method to convince yourselves that the "fundamental kinematic equation" makes sense.  You used the derivative of a polynomial to get the velocity equation from the position equation. You are supposed to instantly recognize the velocity equation that you get as self-evident:

= v(t)

Do the derivatives to get the self-evident equation:

v(t) = ...

(I'm leaving the result out because you're supposed to think it through yourself.)

There's another way to justify the FUNKE (That stands for FUNdamental Kinematic Equation.) Start from the idea that displacement is the area beneath the v(t) graph. Draw the graph for a typical case of constant acceleration and figure out the displacement Δx:

From my Phys100 webpage.

Now I'll colour-code the equation so you can see where each term comes from.

You only need to do some simple substitutions in notation to transform it to the form of the FUNKE at the start:

• Δx = x(t) - x₀
• Δt = t (lazy notation)
• v₁ = v₀

Now our graph that helped us get the equation was just to aid our thinking. It shows a positive a. But nothing in the derivation of the equation assumed that acceleration was positive so the result is general and holds for negative acceleration too.

As an exercise draw the figure for a negative a and see how it works out.

What about negative initial velocity and positive a?

There's one more case. What is it?