Students had used ray tracing to determine image locations for a variety of object locations - concave mirrors produce the biggest variety of image types and locations that we've seen, so each one was different in an important way. They also collected data in the lab using the Pasco mirrors, light source, and screens, and tried to put all of this together into a nice predictive model for these images.
Things have been getting more difficult since we left plane mirrors (which were nice and predictable, with their upright, same size, same distance, virtual images). Convex mirror images were always upright, virtual, and diminished, but the exact location and size have required tracing to determine.
Concave mirrors are even more complex, so after the data was taken, I set the groups off on five tasks, after which they reported back to the class with whiteboards, etc.:
- How do adjustable flashlights work? What's being adjusted, and how does it make the beam go from narrow to parallel to wide?
Here they determined that the bulb position moving from inside the focus to the focus to outside of it was the key.
- What does the image location vs. object location graph look like, and what does the magnification vs. object location graph look like? This'll allow us to see the complicated trend and predict a bit. We actually divided both of the image and object distances by the focal length so that we could use both the ray tracing data and the lab data together on the same graph.
Here they had to think about special-case points (like an object right on the mirror), recognize asymptotes (both vertical - for holes in the function - and horizontal - for long-term behavior), and draw the strangest fit line yet!
- Does this equation that I pulled out of thin air work for the images?
Here they saw that this equation does indeed work, assuming that your signs for all distances are correct. It even works for convex mirrors (where the focal length is negative because the focus is behind the mirror) and for plane mirrors (if you take the focal length to be infinite, you get the equal-but-negative relationship that we've played with all term)!
Even better, this is exactly the kind of algebraic model that we thought that the graph suggested - the vertical asymptote and the flipped left and right sides strongly implied an inverse function, though it is offset, so it wasn't just a simple inverse function. We had the right family, but it's a pretty tricky one to get just from the graph.
- How about these two different definitions of magnification?
Some similar triangles saved the day here!