Quadratics Revisited: The Falling Object Model (part 2)

I worked on this applet for a while, but Frank offered some good pushback which got me to rethink how I wanted to go about this lesson.

Day 1+


Falling Object (Question) from David Cox on Vimeo.

Show the first part of the video and discuss what we notice. I

"9.8 meters per second per second."

I ask, "Alright, so what does that mean?"

"9.8 meters per second per second!"

"So what's the per second per second all about?"

"Hmm."

Show part 2.

Obvious question: "When is it going to hit the ground?"

Pass out the laminated picture and some Vis-a-vis markers (I knew those would come back in style).

"So what do we need to know?"

"The amount of time between each ball."

"The video was exported at 6 frames per second."

"Huh?"

"6 frames per second."

After a brief discussion, we were pretty clear that 6 fps meant that 1/6 second passed between strobes.

Let's go to work

Most kids realize that the strobes' rate of change has a rate of change and they're able to come within a few hundredths of a second from the answer.


Falling Object (Answer) from David Cox on Vimeo.

Day 2+

Project the same still and ask:

"Ok, so how fast is the ball travelling?"

"Huh?"

"How fast is the ball travelling? You guys were talking about 9.8 m/s² yesterday. What's that all about?"

"It's gravity."

"What does that mean?"

We discuss how gravity is an acceleration but we change from metric to ft ('cause that's what's gonna be on THE test) so the 9.8m becomes 32ft.

So how fast is the ball travelling after 1 second? 2 seconds? 3 seconds?

led to

What is the average rate after 1 second? 2 seconds? 3 seconds?

which led to

Then how far has the ball traveled after 1 second? 2 seconds? 3 seconds?

Once we got comfortable with 16t being a rate, it wasn't too far of a stretch to be able to say that 16t² is a distance.

Can we generalize the height with a given starting height (s)?

Day 3+

Now what if I threw the ball down at a rate of 10 ft/s?

The major preconception here was that many students wanted the initial velocity of 10 ft/s to be an acceleration so they just added it to the 16 ft/s². I tried to stay out of it as much as I could and cooler heads prevailed. We eventually decided that the height of a ball after t seconds could be modeled with:

h(t) = -16t² - vt + s

The key to unlocking this function was for each term to be able to stand alone on its own. Once everyone realized that 16t², vt and s are all distances and 16t² has to be negative since the distance has to be subtracted from s in order to get the height, we were good to go.

Now let's have some fun with rational expressions.