What You Will Do:

• Learn how the position of a cart rolling down a ramp can be modeled by a quadratic function.
• Review key concepts: standard form of a quadratic, the parabola it graphs, and how each coefficient shapes that graph.
• Sketch your prediction of what the position-time graph will look like before collecting any data.
• Set up a gentle ramp and collect position-time data as the cart rolls downhill.
• Fit a quadratic model to your data in Desmos and analyze the result.
• Repeat your data collection with a steeper ramp and with a gentle push-start release to see how the quadratic coefficients change.
• Complete the Google Docs 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 cart'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. The graph of a quadratic function is a parabola.
• Parabola: the U-shaped curve that is the graph of a quadratic function. If the coefficient $a$ is positive, the parabola opens upward.
• Coefficient $a$: controls how wide or narrow the parabola is and which way it opens. A larger $\lvert a \rvert$ makes the parabola narrower (the curve rises more steeply); a smaller $\lvert a \rvert$ makes it wider (the curve rises more gradually).
• Coefficients $b$ and $c$: $c$ is the function's value when $T = 0$; this is the cart's starting position. $b$ is the cart's initial velocity - it shifts the curve and affects where its vertex falls.
• Vertex: the highest or lowest point on a parabola, located at $t = -\dfrac{b}{2a}$, where the curve changes from decreasing to increasing (or vice versa). For a cart that starts from rest and only speeds up, the vertex of its position-time graph falls at the start of the motion.
• To review how the coefficients of a quadratic determine the shape and vertex of its parabola, click the video thumbnail below:
  
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• Click this link MotionCart - Activity 2 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?

1. Read the motion description below, then use your mouse to click and drag on the graph to sketch your prediction of what the position-time graph will look like.
   The MotionCart is placed at the top of a gentle ramp and released from rest. It rolls freely down the ramp, speeding up more and more the farther it travels - gravity alone pulls it forward, so its speed keeps changing rather than staying constant.
2. If you need to start over, click Erase Drawing. When you are satisfied with your sketch, click Capture Drawing to copy the image to the clipboard and paste it 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 on a flat surface. 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 gentle ramp using a book, binder, or board propped up at one end. The ramp should be at about 100 cm long (longer is okay) with a rise of roughly 3 to 5 cm - just enough slope for the cart to roll on its own without moving too fast.
5. Place the cart at the top of the ramp with the USB-C port (the front of the cart) facing down the slope, wheels pointing straight down the ramp, so the cart rolls forward as it descends. Hold the cart steady in place with your hand - don't let go yet.
6. Click Zero Position so the starting point reads 0 cm, while still holding the cart in place.
7. Scroll down so the Desmos graph is fully visible before you start collecting.
8. Click Start Collection. The button will turn yellow and display "Waiting for Motion..." - the system is watching for the cart to start moving, but hasn't recorded anything yet.
9. Let go of the cart with a gentle, clean release - don't push or flick it. Any nudge at the start will show up in your data as extra initial speed. As soon as the cart moves, collection begins automatically from position 0 and the button turns red.
10. Collection will stop automatically when the cart comes to rest at the bottom, or you can click Stop Collection to stop early.
11. Use Clear Graph to reset and try again. Repeat until you have a clean run.
12. Once you have a clean run, move on to Analyzing Your Data below to model it.

Analyzing Your Data:

• In the Desmos expression list below, find the Quadratic Model folder and click the circle to its left and then click the triangle to expand the folder and reveal the $a$, $b$, and $c$ sliders.
• Drag the sliders to adjust the model $s(t) = at^{2} + bt + c$ until the curve fits your collected data as closely as possible.
• Record the values of $a$, $b$, and $c$ in your worksheet. Explain what each coefficient does to the graph - what does $a$ tell you about how wide or steep the parabola is? What should $c$ be if the cart started at position zero?
• Now compare your original prediction sketch to your fitted model. How closely does the shape of your prediction match the model once it's adjusted to fit the data? Where are the biggest differences, and what does that tell you about your prediction?
• Find the vertex of your fitted parabola. What does the vertex represent for the cart's motion?
• Discuss whether a steeper ramp would produce a larger or smaller value of $a$, and how the shape of the parabola would change.
• Click the Capture Graph button to copy your graph and then switch to the browser tab with your Google Docs worksheet and paste it into the doc.
• Click Clear Graph, hide the folder by clicking its circle again, then continue on to Going Further below.

Going Further:

• Repeat the collection and analysis with a steeper ramp. Compare the value of $a$ between your steeper-ramp run and your original run. Does the change match the prediction you made in Analyzing Your Data?
• Repeat the collection and analysis adding a little push of the cart during release. Compare the value of $b$ between your push-start run and your rest-released run. How did the push change $b$, and how did it shift the location of the vertex?

Let's Go:

• Click Hide Directions to give yourself more space, then complete the activity and your worksheet.