What You Will Do:
- Explore how the slope and intercept of a linear equation $S = mT + b$ describe the motion of the MotionCart on a position–time graph.
- Work through four parts: discovering slope as speed, intercept as starting position, writing the motion equation from your data, and then predicting and testing a new motion.
- Connect your MotionCart, collect real position–time data, and use Desmos to fit a line and read off the slope and intercept.
- Complete the Google Docs worksheet and submit it according to your teacher's instructions.
- Position–time graph: a graph where the horizontal axis is time $T$ (seconds) and the vertical axis is the cart's position $S$ (centimeters from its zero point).
- Linear equation of motion: when the cart rolls at a steady speed, its position-time graph is a straight line described by:
$$S = mT + b$$where $m$ is the slope and $b$ is the $S$-intercept.
- Slope $m$: the slope of the position–time line equals the cart's velocity - how fast it moves and in which direction.
- $m = \frac{\Delta S}{\Delta T}$ - rise over run on the graph, in cm/s.
- A positive slope means the cart moves forward (position increasing).
- A negative slope means the cart moves backward (position decreasing).
- A larger $|m|$ means the cart moves faster.
- Intercept $b$: the value of $S$ when $T = 0$ - the cart's starting position at the moment collection begins. If you zero the cart and start collecting immediately, $b \approx 0$. If you start the cart already displaced from zero, $b$ will reflect that offset.
- To review slope and intercept on position–time graphs before starting, click the video thumbnail below:
- Click this link MotionCart - Activity 1 to open the worksheet in a new browser tab. Click Make a copy to save your version to your Google Drive.
- For detailed setup instructions and how to use your MotionCart, complete the Getting Started with MotionCart activity first.
- Place a small piece of tape in the center of your table and mark it with a zero. This is the cart's starting position.
- Place the front wheels (the USB-C port is at the front of the cart) on the zero mark, turn on your MotionCart and connect it. Roll the cart to the right and ensure the position of the cart is positive and increasing. If it is backward, click the Change Sign button.
- Return the cart to the zero mark and click the Zero Position button.
- Click Start Collection and roll the cart forward at a slow, steady speed for 10 seconds. You may want to clear the graph to try rolling it smoothly a few times until you get the hang of it.
- In the Desmos window below, display the Part A folder (click on the circle to the left) and expand the folder (click the triangle) - Read the slope $m$ from the Desmos expression list and record it in your worksheet.
- 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. Place your cart back on the piece of tape (zero position) and click the Zero Position button. Now roll the cart forward again at a faster, steady speed. Repeat the regression and record the new slope. Compare the two slopes - which slope is greater?
- Try a roll in the opposite direction. What is the sign of the slope? Record your observation in your worksheet.
- Click Clear Graph. Place the cart at the zero mark and click Zero Position. Click Start Collection, roll the cart forward at a steady speed for 10 seconds. Record the intercept $b$ from the regression line in Desmos. It should be close to 0 cm.
- Click Clear Graph. Now roll the cart 20 cm forward of the zero mark (do not click Zero Position); the red initial position should be at the 20 mark on the Y-axis. Click Start Collection, roll forward again for 10 seconds. Record the new intercept. What does it represent?
- Click Clear Graph. Roll the cart 20 cm behind the zero mark. Repeat the collection. Record the intercept. Is it positive or negative? Why?
- 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.
- Return your cart to your table's zero position. Click Clear Graph and Zero Position. Turn off the part 1 and 2 folder and turn on the part 3 folder.
- Design a roll: choose a starting position (e.g., 50 cm behind zero) and a target speed (e.g., 5 cm/s forward). Adjust the slope (m) and intercept (b) sliders to your predicted equation $S = mT + b$
- Roll the cart to your chosen starting position (do not click Zero Position - that would set $b = 0$). Click Start Collection, roll at your target speed for 10 seconds.
- Compare the measured position with the predicted position from the equation.
- Expand the data table and find and record the measured position (S) at T = 5 seconds.
- Click on the blue dashed line in the desmos window at time = 5 seconds and record the predicted position from the equation.
- Determine how close the two values are by calculating the percent error between the predicted and measured positions.
- Click Capture Graph and paste the result into your worksheet.
- Your teacher will give you a target equation, for example $S = -4T + 30$. Consider how the slope and intercept of the equation inform the motion of your cart.
- On the graph below, sketch what you think the graph of the target equation will look like before collecting any data. When finished, click Capture Drawing and copy and paste it into your worksheet.
- Adjust the slope (m) and intercept (b) sliders to match the target equation $S = mT + b$
- Set your cart on the zero position and click the Zero Position button. Roll your cart to the position indicated by the intercept. Click Start Collection, roll at the speed and direction indicated by the slope for 10 seconds.
- In the Desmos expression list, open the "Going Further" folder. Use the $m$ and $b$ sliders to set a target line, then challenge a partner to replicate it exactly with their cart. Swap roles and repeat.
- Click Hide Directions to give yourself more space. Work through Parts A–D in order, capturing and pasting each graph into your worksheet before moving on.
$$\% \text{ error} = \frac{|\text{predicted} - \text{measured}|}{|\text{measured}|} \times 100\%$$