Confusing moments in physics instruction.

This past Friday I tried to compute the distance an object had traveled while being constantly accelerated (decelerated) as

D = V * T [WRONG!]

My answer differed from the solution manual by a factor of (0.5) which had me stumped in front of my students. As I pondered my confusion both then and now, I realized something sublime and beautiful.

Distance traveled with no acceleration: D = V * T

Distance traveled in the presence of constant acceleration D = (0.5) * (Vi + Vf) * T [which we have been calling the Trapezoid Rule in class.]

The sublime part is that I had honestly never really contemplated the similarity between the two equations especially when Vi (initial velocity) or Vf (final velocity) equals zero. The beautiful part is that no matter what the value of the acceleration is, you are constrained from how far you can travel by the factor of (0.5).

Now the debate in my head is, do I point this error / confusion out to students, will they appreciate or grow from it? Does this type of vulnerability (that your teacher sometimes gets confused, but presses on, and finds new wonderments) really help? I know that this type of stimulation for me is a key benefit which I see from teaching.

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