Unit 4: Considering the Influence of Light and Thermal Phenomena on Global Clima
Unit 4: Considering the Influence of Light and Thermal Phenomena on Global Climate
VII. Developing Mathematical Representations of Changing Quantities
The world is full of changing quantities. Key to becoming aware of what is happening is measuring how much a quantity is changing. That knowledge can provide the basis for thinking about what might be happening next and perhaps for perceiving ways to increase positive outcomes or at least minimize negative ones.
This section develops a mathematical way of representing changing quantities. The first activity explores changing quantities in an everyday context, a tossed ball. The next presents evidence for a changing quantity relevant to global climate change and its impact, estimates of how fast glaciers are melting. Development of an analogy between motion and melting graphs guides making projections for how fast glacial ice may be melting in the future.
A. Developing familiarity with motion graphs for a tossed ball
Rolling, kicking, throwing, and catching a ball are familiar activities for most people. To describe mathematically what is happening to the ball, however, one needs some technical skills: ways to label where the ball is, its position, to describe how that position is changing, the ball’s velocity, and to think about how that velocity is changing, its acceleration. By looking at graphs of how a ball’s position, velocity, and acceleration are changing with time, one can learn a lot about a ball’s motion – is it at rest? Moving with constant speed? Speeding up? Slowing down? Similarly, one can learn a lot about how a quantity of interest is changing by looking at analogous graphs for that quantity, such as how fast are the world’s glaciers melting?
Question 4.14 How do position, velocity, and acceleration of a tossed ball change with time?
To explore this question, use:
- whiteboard,
- basketball,
- motion detector connected to a computer or graphing calculator
- If a motion detector is not available, use:
- Basketball
- Stopwatch,
- Flat board,
- Meter stick or yard stick
- Book or wood block to incline the flat board
A motion detector works by emitting ultrasound waves that are reflected from an object back to the detector. The detector reports how far away the object is and makes a graph of position (where the object is) versus time (the instant it was there). We use a Go Motion! detector that connects directly to a computer (https://www.vernier.com/products/sensors/motion-detectors/).
- Start by exploring what a motion looks like on a position versus time graph (x vs t) with the vertical axis representing where the object is (x, its position with respect to the motion detector’s position), and the horizontal axis representing the time the object was there (t, a clock reading). What can you find out about position vs. time graphs by standing still as well as by moving back and forth in front of the motion detector?
(If you do not have access to a motion detector, lay a meter or yard s