Preliminary In the function p(t) = p0 + vt does the product vt mean v added together t times, or t added together v times?

Motion II

Accelerated Motion Finger tool Position Time Position and Time Speed Speed and Time How would you describe the motion of the cart? Acknowledge that many of the teachers may know several things about the mathematics and physics of this situation. Emphasize that they will need to focus on seeing the connections of the major ideas across all representations, and that there are important but subtle ideas which can come out of doing this. For p vs. t, ask whether they are moving their finger faster because the car is moving faster. Does this mean you are thinking about position as the quantity? Try again trying to focus on position as the quantity.

Connect your calculator to the CBR with the 1/8” stereo jack cable: Press [APPS] on the calculator and scroll down to choose CBL/CBR. At the main menu, choose Ranger. Then choose SETUP/SAMPLE. Use the arrow keys ◄►▲▼ and [ENTER] to change the options as below. You may use either meters or feet for the units. MAIN MENU START NOW REALTIME: YES TIME (S): 15 DISPLAY: DIST BEGIN ON: [ENTER] SMOOTHING: NONE UNITS: METERS Cursor up to “START NOW” and press [ENTER], then press [ENTER] again when prompted to begin the data collection.

Collect your own data: Start your cart Place the detector At the top of the ramp or at the bottom Moving fast or moving slow Place the detector It is best to have the detector about ½ meter away from the ramp since it does not detect close objects accurately. Once teachers have collected their data, have them import it into Fathom (instructions are in the Fathom How-To) document.

Analyze your data in Fathom: A complete analysis of position vs. time (table, graph, formula with sliders, units) Compute the average velocity (V) between each data point A complete analysis of velocity vs. time (table, graph, formula with sliders, units) Compute the average acceleration (a) between each data point A complete analysis of acceleration vs. time (table, graph, formula with sliders, units) Teachers may struggle to know what to do in each of these analyses.

Discussion What connections did you see in your three analyses: position, velocity, acceleration? Can you see these connections in other representations? What connections did you see across representations? Push for them to see any connection in each representation. If they miss a representation in their description, ask how you could see that connection graphically, for example.

Average Velocity Formula: Graph: Context? p t position time Emphasize that the secant line is ANOTHER LINE – with what key properties? What does that line correspond to? What would it mean if it were the graph for an actual motion?

Average Rate Slope of the secant: The constant slope of an auxiliary line that gives the same amount of change in height (“rise”) over the same change horizontally (“run”). Average velocity: The constant velocity for an auxiliary trip that gives the same change in position during the same time interval. Average rate: The constant rate for an auxiliary situation that results in the same amounts of change in both quantities. Emphasize that average rate is the general version. Any average rate can be understood in this way. The slope of the secant and average velocity are just two examples of this.