Post Calculus Exam: Sonic Boom

I spent 2 of the class periods after the AP Exam exploring and expanding on a cool problem I found in Smith and Minton. The problem is about what a sonic boom is, and how you can visualize it graphically.

The first day we had to refresh our memories on parametric equations and graphing. Then the students looked up various things on the internet and wrote it in their notebooks as such:

We had some good discussion on the various speeds and if they are "exact" and what could change the values. Then I made sure they got what a Mach number is by the following "quiz" in their notebooks:

I had them look up the speeds of various vehicles and we tried to get a wide range of numbers and then they had to figure out the Mach numbers based on their speed of sound.

On the 2nd day we then read over the "yellow" paper from page one, and I had to make sure they understood what a unit on the graph was. That was fascinating. It was more of a struggle for them than I would have thought, but we eventually got to it, and again, good discussion.

Then the actual graphing began. We had to take it slowly and keep talking about what the units meant and how you made decisions and such. Here is the graph for a jet going Mach 0.8. We had to keep stressing that the "arcs" were at time 5 seconds after "start" of the 5 noises the jet made at various times:

 

Then we graphed the "same thing" but for the jet moving at Mach 1:

 

 The students were excitedly asking me question upon question, because really it is cool and wonder-ful ... and I kept mentioning that I'm not an expert, but that we should write their questions down to think about for later.

Then finally we did the same process for a jet moving FASTER than the speed of sound ... Mach 1.4:

 

We had to be cautious with scales and such and all sorts of great mathy things came up. Then class was over. 

I loved their enthusiasm and questions and wonderment at this actual topic.