Linear motion (Calculus)
Explore the premise of classic AP Calculus AB prompt: "A particle moves along the x-axis..."
In the GeoGebra construction below, you're provided an [color=#999999]x(t)[/color] curve that models such particle linear motion, with the horizontal axis representing time t and the vertical axis representing position x. Drag the "flip/shrink/flat" slider to transform the curve so that it lies flattened along the x-axis. [br]Now you also see a [color=#0000ff]blue curve[/color] that represents the same particle motion, but in this case the horizontal axis represents position x. I recommend stopping the slider just shy of the end so that the curve is not completely flat, and you can better see the left/right motion of the [color=#0000ff]blue particle/point[/color].[br][br]To get the particle moving, either: (1) Press the "start" button, (2) Drag the "t" slider, or (3) Drag the [color=#b6b6b6]grey point[/color] along the x(t) curve. Study how the motion of the [color=#b6b6b6]grey point[/color] on the x(t) curve correlates to the motion of the [color=#0000ff]blue point[/color] along the horizontal x-axis.
[list][*]When the [color=#b6b6b6]grey x(t) point[/color] is increasing, the [color=#0000ff]blue point[/color] is moving _______________.[br][/*][*]When the [color=#b6b6b6]grey x(t) point[/color] is decreasing, the [color=#0000ff]blue point[/color] is moving _______________.[/*][*]When the [color=#0000ff]blue point[/color] changes directions, the [color=#b6b6b6]grey x(t) point[/color] is at a __________ or __________.[/*][/list]
Check the "[color=#ff0000]show x'(t)[/color]" box to display the velocity curve. Optionally, you may want to uncheck the "[color=#999999]show x(t)[/color]" box to hide the position curve and reduce screen clutter. If you haven't already started the particle animation, press the "start" button. Study how the motion of the [color=#ff0000]red point[/color] on the [color=#ff0000]x'(t)[/color] curve correlates to the motion of the [color=#0000ff]blue point[/color] along the horizontal x-axis. Alter the "speed" slider to change how fast the animation plays.
[list][*]When the [color=#ff0000]red point[/color] is above the horizontal t-axis, the [color=#0000ff]blue point[/color] is moving _______________.[/*][*]When the [color=#ff0000]red point[/color] is below the horizontal t-axis, the [color=#0000ff]blue point[/color] is moving _______________.[/*][*]When the [color=#ff0000]red point[/color] reaches a local minimum or maximum on the [color=#ff0000]x'(t)[/color] curve, the [color=#0000ff]blue point[/color] ____________________.[/*][/list]
Check "extra settings":[br]The "degree" slider allows [color=#999999]x(t)[/color] to be a polynomial of different degree. As long as "[color=#b6b6b6]show x(t)[/color]" is also checked, you will see [color=#b6b6b6]grey Xs[/color] in the graphics region. Drag the [color=#b6b6b6]grey Xs[/color] up/down to change the shape of the [color=#b6b6b6]x(t)[/color] curve. Drag the "spacing" slider to alter how far apart horizontally the [color=#b6b6b6]grey Xs[/color] are.[br]Drag the "shrink-to-flat" slider to alter how high above/below the horizontal axis the [color=#0000ff]blue curve[/color] is before it falls completely flat when dragging to the end of the "flip/shrink/flat" slider.