What You Will Do:

• Build a ramp and collect position-time data as the MotionCart rolls down the ramp.
• Part 1 - Modeling: Fit a quadratic model, $s(t) = at^{2} + bt + c$, to your collected data using sliders in Desmos.
• Part 2 - Secant: Use a secant-line tool to estimate the average rate of change between two points on your fitted curve, then shrink the interval $h$ toward zero and compare the result to the measured velocity (V).
• Part 3 - Derivative: Find the value of your model's symbolic derivative at that same point and compare it to your Part 2 result - connecting different ways of finding the same instantaneous rate of change.
• Complete the Google Docs mc_5 worksheet and submit it according to your teacher's instructions.

Getting Ready:

• Position-time graph: a graph where the horizontal axis shows time (seconds) and the vertical axis shows the MotionCart's position (centimeters from its zero point).
• Quadratic function: a function of the form $f(t) = at^{2} + bt + c$, where $a$, $b$, and $c$ are constants. Its graph is a parabola. A cart rolling freely down a ramp speeds up as it goes, so its position over time follows a quadratic function.
• Secant line: a line connecting two points on a curve. Its slope equals the average rate of change of the function between those two points:
  $$m_{sec} = \frac{\Delta S}{\Delta T} = \frac{S(t_{2}) - S(t_{1})}{t_{2} - t_{1}}$$
• Average rate of change: how much position changes, on average, per unit of time over an interval. On a position-time graph this is a velocity in cm/s - and it is exactly the slope of the secant line connecting the interval's endpoints.
• Instantaneous rate of change: the rate of change at a single instant, rather than averaged over an interval. As the time interval $\Delta t$ shrinks toward zero, the secant slope approaches the slope of the tangent line at that point. This limiting value is called the derivative:
  $$\frac{dS}{dT} = \lim_{\Delta T \to 0} \frac{\Delta S}{\Delta T}$$
• Tangent line: the line that touches a curve at exactly one point and shares the curve's slope at that point. The slope of the tangent line is the derivative at that point.
• Derivative of a quadratic: for a position function $s(t) = at^{2} + bt + c$, the derivative (using the power rule) is:
  $$\frac{d}{dt}\left(at^{2} + bt + c\right) = 2at + b$$
  Notice that this derivative is itself a linear function of $t$ - this is why constant acceleration produces a straight-line velocity-time graph, as you saw in Activity 2.
• Tools in this activity's Desmos graph:
  • Quadratic Model - Position folder: sliders $a_{1}$, $b_{1}$, $c_{1}$ for the curve $f(x) = a_{1}x^{2} + b_{1}x + c_{1}$ you will fit to your data.
  • Secant folder: sliders $x_{0}$ (the point on the curve you're studying) and $h$ (the size of the interval). It plots the two points $p_{1} = (x_{0}, f(x_{0}))$ and $p_{2} = (x_{0} + h, f(x_{0} + h))$, the line through them, and computes the secant slope:
    $$m_{secant} = \frac{f(x_{0} + h) - f(x_{0})}{h}$$
    This is exactly the difference quotient - the formula behind the formal definition of the derivative.
  • Derivative folder: the symbolic derivative of your fitted model, which you can evaluate at $x_{0}$ to check against your secant-slope result.
• Click this link MotionCart - Activity 7 to open the worksheet in a new browser tab. Click Make a copy to save your version to your Google Drive.

What Do You Think? - Part 1: Modeling:

1. Recall: a quadratic function has the form $f(t) = at^{2} + bt + c$, and its graph is a parabola.
2. Read the motion description below.
   The MotionCart is placed at the top of a gentle ramp and released from rest. It rolls freely down the ramp, accelerating steadily. No one pushes it - gravity alone pulls it forward.
3. Using Draw (red) sketch the curve, Point (black) mark three points along it, and Line (blue) draw a tangent at each; then Capture Drawing and paste it into your worksheet.
Mode: Draw Capture Drawing Erase Drawing

What Do You Think? - Part 2: Secant:

1. Recall: a secant line connects two points on a curve, and its slope is the average rate of change between them, $m_{sec} = \frac{\Delta S}{\Delta T}$.
2. To review secant lines and average rate of change, click the video thumbnail below:
   
   ×
3. On the graph below (a parabola), click once at $T = 0.4$ to place the first point, then move your mouse toward $T = 1.4$ and click again to place the second point and draw the secant line between them.
4. Calculate the slope of the secant line you just drew and report it in your worksheet.
5. Repeat steps 3 and 4 at the points $T = 0.4$ and $T = 0.5$
6. If you need to start over, click Erase Drawing (a third click on the graph will also start a new line). When you are satisfied, click Capture Drawing and paste the graph into your worksheet.
Capture Drawing Erase Drawing

What Do You Think? - Part 3: Derivative:

1. Recall: the derivative of a function gives its instantaneous rate of change - the slope of the tangent line - at every point.
2. To review why we study derivatives, click the video thumbnail below:
   
   ×
3. The graph below shows the same parabola from Part 2. Sketch your prediction of what the derivative of this graph looks like - not the parabola itself, but the curve formed by its slope at every point.
4. If you need to start over, click Erase Drawing. When you are satisfied, click Capture Drawing and paste the graph into your worksheet.
Capture Drawing Erase Drawing

Using Your MotionCart:

1. For detailed setup instructions, complete the Getting Started with MotionCart activity first.
2. Turn on your MotionCart, connect it, and click Zero Position.
3. Practice rolling the cart forward and back. Confirm that position increases when rolling forward and decreases when rolling back. If the sign is reversed, click Change Sign and roll again to confirm.
4. Build a ramp using a board, table leaf, or similar - about 2 meters long propped up at roughly a 20° incline works well for a clear, fast quadratic run. A shorter or gentler ramp will also work, but produces a less pronounced curve.
5. Place the cart at the top of the ramp with its wheels pointing straight down the slope. Click Zero Position so the starting point reads 0 cm.
6. Scroll down so the Desmos graph is fully visible.
7. Click Start Collection. The button will turn yellow and display "Waiting for Motion..." while the system waits for the cart to begin rolling.
8. Release the cart and let it roll freely down the ramp. Do not push it. Once the cart moves, data collection begins automatically and the button turns red.
9. Collection will stop automatically when the cart comes to rest at the bottom, or you can click Stop Collection to stop early.
10. Use Clear Graph to reset and try again. Repeat until you have a clean run, then click Capture Graph and paste the result into your worksheet.

Analyzing Your Data - Part 1: Modeling:

• Compare your prediction sketch to the collected position-time graph. Describe any differences in your worksheet.
• In the Desmos expression list, click the folder icon next to Quadratic Model - Position to expand and display it.
• Adjust the sliders $a$, $b$, and $c$ to fit only the accelerating portion of your data - ignore the flat, negative-time data before release, and ignore any later data where the cart slows after running off the end of the ramp.
• Record the values of $a$, $b$, and $c$ in your worksheet.

Analyzing Your Data - Part 2: Secant:

• Expand and display the Secant folder. Set $T_{0}$ to a time partway through your run (for example, $T_{0} = 0.25$).
• Drag the $h$ slider from large down toward its smallest value. Watch the points $p_{1}$ and $p_{2}$ move closer together and the secant line rotate as $m_{secant}$ updates.
• Record $m_{secant}$ for a few decreasing values of $h$ in your worksheet.
• State the value $m_{secant}$ appears to be approaching as $h \to 0$.
• Find the value of the V column in the data table at (or very near) $T_{0}$. Compare it to your $m_{secant}$ result as $h \to 0$ and explain why they should be close.

Analyzing Your Data - Part 3: Derivative:

• Expand the Derivative folder. This plots the symbolic derivative of your fitted model.
• Compare it to your prediction sketch from What Do You Think? - Part 3. Describe any differences in your worksheet.
• Find the value of the derivative at $T_{0}$ (the same point you used in Part 2) and compare it to your $m_{secant}$ result as $h \to 0$.
• Explain the connection between the secant slope as $h$ shrinks toward zero, the derivative, and the velocity you measured.
• Move $T_{0}$ to a different point on your curve and repeat Parts 2 and 3. State whether the derivative's value still matches V at that new point.
• When finished, submit your worksheet according to your teacher's directions.

Going Further:

• Try releasing the cart with a gentle push instead of from rest, collect new data, and refit your quadratic model. Which coefficient changed the most - $a_{1}$, $b_{1}$, or $c_{1}$? Set $T_{0}$ near the start of this new run and check whether $m_{secant}$ as $h \to 0$ still matches V there.
• Try a much steeper or much gentler ramp angle. How does the shape of your fitted curve change, and how does that show up in the Secant and Derivative tools?

Let's Go:

• Click Hide Directions to give yourself more space. Build your ramp, place the cart at the top, click Zero Position, then click Start Collection and release the cart. After your run, work through Parts 1, 2, and 3 of the analysis.