What You Will Do:

• Learn how quadratic functions describe motion with constant acceleration, such as a cart rolling down a ramp.
• Review key concepts: how a quadratic function models position over time when acceleration is constant, and how the coefficients relate to the physical motion.
• Sketch your prediction of what the position-time graph will look like before collecting any data.
• Connect your MotionCart, 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.
• Complete the challenge graph-matches by adjusting your ramp to produce different quadratic curves.
• 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).
• Acceleration: the rate at which velocity changes over time. When acceleration is constant, velocity increases (or decreases) at a steady rate. A cart rolling down a ramp experiences approximately constant acceleration due to gravity.
  $$a = \frac{\Delta v}{\Delta t}$$
• 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. When an object moves with constant acceleration, its position over time follows a quadratic function:
  $$s(t) = \tfrac{1}{2}at^{2} + v_{0}t + s_{0}$$
  Here $a$ is the acceleration, $v_{0}$ is the initial velocity, and $s_{0}$ is the starting position. For a cart released from rest on a ramp ($v_{0} = 0$, $s_{0} = 0$), this simplifies to:
  $$s(t) = \tfrac{1}{2}at^{2}$$
• Parabola: the U-shaped curve that is the graph of a quadratic function. If the coefficient of $t^{2}$ is positive, the parabola opens upward - the cart's position increases faster and faster as it rolls down the ramp.
• Coefficient $a$: in $s(t) = \tfrac{1}{2}at^{2}$, the value of $a$ controls how quickly the curve bends upward. A steeper ramp produces a larger acceleration, which means a wider, faster-growing parabola. A gentler ramp produces a smaller acceleration and a narrower, slower-growing curve.
• Velocity-time graph: when acceleration is constant, the velocity-time graph is a straight line. The slope of that line equals the acceleration. If the cart starts from rest, the line begins at zero and rises steadily.
• To review acceleration concepts before starting, 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, accelerating steadily. No one pushes it - gravity alone pulls it forward.
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. 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 about 50 to 100 cm long 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 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.
11. There are challenge graphs in the Desmos expression list. Click the folder icon next to Challenge 1 to display the first target curve. Adjust your ramp angle and repeat the collection process - click Start Collection, release the cart, and try to match the target curve.
12. When you are satisfied with your match, click Capture Graph and paste the result into your worksheet. Then click Clear Graph, hide the completed challenge by clicking its folder icon, and display the next one.
13. Repeat until you have completed and pasted all challenge graphs into your worksheet.

Analyzing Your Data:

• Compare your prediction sketch to the collected position-time graph. Describe any differences in your worksheet.
• Open the quadratic model folder in the Desmos expression list. Adjust the sliders to fit the curve $s(t) = at^{2} + bt + c$ to your data as closely as possible.
• Record the values of $a$, $b$, and $c$ in your worksheet. Explain what each coefficient represents physically - what does $a$ tell you about the acceleration? What should $b$ and $c$ be if the cart started from rest at position zero?
• Look at the velocity-time graph (the V column). Is it approximately a straight line? Explain why constant acceleration produces a linear velocity-time graph and a quadratic position-time graph.
• Discuss whether a steeper ramp would produce a larger or smaller value of $a$, and how the shape of the parabola would change.
• Complete the challenge graph-matches and paste each captured graph into your worksheet.
• When finished, submit your worksheet according to your teacher's directions.

Going Further:

• Try changing the ramp angle and collecting new data. Compare the value of $a$ from your quadratic fit for the steeper ramp to the original. Does it match your prediction?
• Try releasing the cart with a gentle push instead of from rest. How does the graph change? What happens to the value of $b$?

Let's Go:

• Click Hide Directions to give yourself more space. Set up your ramp, place the cart at the top, click Zero Position, then click Start Collection and release the cart. After your initial run, work through the challenge graphs.