The Rock Problem

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Objective

SWBAT relate first and second derivatives to velocity and acceleration.

Big Idea

Slopes of tangent lines take on new meaning when given in the context of a falling rock.

Launch and Explore

25 minutes

I love today's lesson! First and second derivatives can be very abstract to students and today we give them meaning and students are presented with a context that make them seem very real. Today's lesson start by giving students this worksheet and having them work on it with their table group. It is rather lengthy, so I want to give them plenty of time to work on it and digest the concepts.

While they are working, there are a few key ideas that I watch out for. I want to make sure that students are on the right track and are not getting tripped up by simple ideas. Here are a few things that I monitor:

  • Average velocity: For questions #1-5, student must find the average velocity over a specific interval. If students get stuck, I ask them how we measure velocity to remind them that we just calculate distance divided by time. 
  • Understanding instantaneous velocity: Students may not understand this concept, so I liken it to when a police officer clocks someone using a radar gun - it is the speed at precisely one instant (theoretically).
  • Calculating the instantaneous velocity: I don't expect all students to figure out #6, but I at least hope they can see that as the interval gets smaller, we are approaching a better estimate to the instantaneous velocity.
  • Finding the velocity at the highest point: It may seem strange to students that velocity can be zero when the rock is at the highest point, but the rock must stop at some point when switching directions.
  • The first derivative: If students do not pick up on the fact that the first derivative is giving velocity, then I ask them to decide what unit the function is measured in to see if that will help.

Share

20 minutes

To being our discussion we go through our answers to questions #1-5 from the worksheet and I choose one student to explain their process to the class. I make sure that during our discussion, students understand that we really used the slope formula to figure out the average velocity. When it comes time to answer #6, it will hopefully be clear that we need to find the slope of the tangent line at t = 1.

The questions on the back are a little more involved. In the videos below I highlight some key points and suggestions I make when discussing some of these questions.

#8 - The derivative of the position function:

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 #9 - The second derivative of the position function:

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#10 - Finding the time when the speed of the rock is 30 feet per second: Make sure that students realize that the speed is 30 feet/sec when the velocity is 30 feet/sec or -30 feet/sec. Thus, the rock will hit 30 feet/sec on the way up and on the way down.


Summarize

5 minutes

This lesson is always demanding, but I find that my students seem to be really engaged and we have some great discussions. To close the lesson, I ask students to think about the following to summarize everything that we learned:

  • How are the position function, velocity function, speed function and acceleration function related?
  • For each of the position, velocity, and acceleration functions, what is x-axis measuring? The y-axis?

Finally, here is a homework assignment to reinforce the work we did with velocity and acceleration.