To understand how the speed of a vector valued function of one variable is defined.
To understand how different vector valued functions can trace the same curve in space but have different motion along the curve.
Explore relationship between the speed and the distance covered over a fixed time.
Figure5.3.1.A screenshot from the lesson regarding the speed of a vector valued function.
This lesson defines and explains the measurement of speed (as a scalar). Several different vector valued functions are given where the same helix graph is traced out, but the motion along the curve (and thus the speed) is quite different across the different functions. The conceptual notion of speed as being distance traveled over time is explored by a few multiple choice questions. All four of the base examples used in the vector valued functions of one variable lessons are animated and available for exploration at the end of the lesson.
Subsection5.3.1Speed Explorations
Exploration5.3.1.
In this exploration, we want to examine whether speed depends on a particular parameterization used or just the location on the curve at which the measurement is taken.
To get started, make sure your headset and controller is set up with the CalcVR app. Start the CalcVR app go to the Vector Valued Functions set of lessons. Start the Vector Valued Function Demo and put your phone in your headset. Once the Demo loads, a 3D graph of a curve in space should generate in front of you with a plot of related scalar quantities to the right. The function input panel will show up on the left of the 3D plot and an option selector panel will show up below the 3D plot.
You should use the option selector panel to turn off all vectors except for the position and velocity vectors. On the function input panel, select the Preloaded Inputs button and scroll until you reach Exploration 1. Click the Display button to load these functions. If a new plot does not automatically generate, then click on the Plot Graph button to the bottom right of the function input panel. You can use the slider at the top of the option selector panel to move the car to different spots on the curve. Note that as you change the car’s location, the \(t\)-value of the location is updated and the corresponding plot is highlighted on the 2D plot to the right.
(a)
Sketch the graph of the speed and note the scale on both axes. Be sure to use the scale on the left of the plot (in green).
(b)
Select the Preloaded Inputs button at the bottom of the Function Input Panel. Scroll to the Exploration 2 functions and select the Display button. The same curve as was used earlier should be rendered but the parameterization used moves through these points twice as fast in terms of t. Note that this means half of the range of t. You should use the Option Selector Panel to turn off all vectors except for the position and velocity vectors.
Draw the graph of the speed for the parameterization of Exploration 2 and be sure to note the scale for both the vertical and horizontal axes.
(c)
Write a paragraph that answers the following questions: Is the shape of the speed graph different? What changed for the values of the speed? What changed on the horizontal scale?
(d)
Select the Preloaded Inputs button at the bottom of the Function Input Panel. Scroll to the Exploration 3 functions and select the Display button. The same curve as was used earlier should be rendered but the parameterization used moves through these points exponentially fast. You should use the Option Selector Panel to turn off all vectors except for the position and velocity vectors.
Draw the graph of the speed for the parameterization of Exploration 3 and be sure to note the scale for both the vertical and horizontal axes.
(e)
Write a paragraph that answers the following questions: Is the shape of the speed graph different? What changed for the values of the speed? What changed on the horizontal scale?
(f)
Based on your work with these functions, write a few sentences about how you know that speed is a property of the driver and not a property of the road.
