We've been working on related rates in calculus. One of the "classic" calculus problems involves a ladder in motion. Its typically moves away from a wall at a constant rate, and the students are asked to determine how fast the top of the ladder is falling.
At least the problem has context, even if the constant rate bit is a stretch. Last year I tried to turn this problem into reality, and it didn't quite work out. This year we did much better. Key things:
-put wheels on the top of the ladder so it rolls smoothly down the wall
-rest the bottom of the ladder on a constant velocity buggy (borrow one from the physics teacher)
-use a good tripod
Here's a snapshot of our setup (screenshot from LoggerPro):
I scaled the video, set up a coordinate system, and tracked the bottom of the meterstick (using the rearmost wheel on the vehicle). This produced the graph of the horizontal position of the bottom of the meterstick below.
I added a trendline so we could get velocity, and then I asked the students to use the length of the meterstick/wheel contraption along with this velocity to predict how rapidly the top of the meterstick would be falling when the base was 32 cm away from the wall (hence the cluster of points near that position).
This is where related rates came in, and the students ended up with an answer. We confirmed it by analyzing the same video and tracking the top of the meterstick.
We got a really nice parabolic shape, and the instantaneous velocity of the top of the meterstick matched their prediction. It was pretty successful- as one student put it, "it's so nice to use math to analyze something complicated that happens in the real world!"
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