What You Will Do:
- Sketch your prediction of what a graph of distance from a landmark vs. time will look like as the MotionCart rolls once past a fixed point on the track.
- Collect position vs. time data in the Desmos Graphing Calculator by rolling the MotionCart in a single, steady pass past a landmark you choose and measure ahead of time.
- Use your measured landmark position to compute distance from the landmark at every moment, and find the graph's vertex.
- Use sliders to fit an absolute value model to your data and compare the fitted vertex to your measured landmark.
- Complete the Google Docs worksheet and submit it according to your teacher's instructions.
- Position (S): how far the cart has moved from its zero position, measured in centimeters (cm).
- Time (T): how many seconds have passed since the cart started moving at the beginning of collection, measured in seconds (s).
- Landmark: a fixed reference point on the track - separate from the cart's zero position - that you measure ahead of time and mark (for example, with a piece of tape). In this activity the landmark is the point the cart rolls past.
- Absolute value: the non-negative distance a number is from 0, written $|x|$. It's defined piecewise:
$$|x| = \begin{cases} x & x \ge 0 \\ -x & x < 0 \end{cases}$$Whichever sign x has, $|x|$ flips it to positive - or leaves it at 0 if x is already 0.
- Vertex form of an absolute value function: $y = |x - h| + k$, whose graph is a V-shape with its corner - its vertex - at the point $(h, k)$. Changing h slides the V left or right; changing k slides it up or down.
- Distance from a landmark: if H is your landmark's measured position, the cart's distance from it at any moment is $D = |S - H|$ - always non-negative, whether the cart is on the near side or the far side of the landmark.
- To review why $|a - b|$ gives the distance between two numbers - the same idea behind $D = |S - H|$ above - click the video thumbnail below:
- Click this link MotionCart - Activity 3 to open the worksheet in a new browser tab. Click Make a copy to save your version to your Google Drive.
- As the MotionCart rolls past a fixed landmark in one steady pass, its raw position S increases the whole time - there's nothing V-shaped about position vs. time by itself.
- But its distance from the landmark, $D = |S - H|$, behaves differently: D shrinks as the cart approaches the landmark, bottoms out right as the cart passes it, then grows again as the cart continues on. Graphed against time, that's a V - the same shape as $y = |x|$, just shifted so its vertex sits at the landmark instead of at the origin.
- The time at which the cart is closest to the landmark becomes h in the vertex form, and however close the cart actually got (rarely exactly 0, since position is only sampled at discrete instants) becomes k. Your job is to find that vertex in your own data and fit $y = |x - h| + k$ to it.
- Picture the MotionCart rolling in a straight line, once, past a landmark partway down the track. Sketch your prediction of what a graph of the cart's distance from the landmark vs. time will look like over that roll. If you need to start over, click Erase Drawing. When you are satisfied, click Capture Drawing to copy the image to the clipboard and paste it into your worksheet.
- During collection, position is recorded in the S column of the Desmos data table alongside elapsed time in the T column.
- Choose a landmark point somewhere in the middle of your track - not right at either end - and mark it (a small piece of tape works well).
- Press and hold the button on the MotionCart for one second so the blue LED blinks, then click Connect Cart in the top bar and pair your MotionCart.
- Place the cart at the very start of the track and click Zero Position so Position reads 0.00 cm.
- Roll the cart forward just until it reaches the landmark, and record the Position reading shown in the top bar - this is your landmark's measured position, H. Then roll the cart back to the start of the track.
- Click Zero Position again at the start of the track to make sure you're beginning from exactly 0.
- Click Start Collection, then roll the cart in one steady pass all the way past the landmark to the far end of the track, without reversing direction.
- Collection will stop automatically when the cart is held still for a few seconds, or you can click Stop Collection to stop early.
- Repeat collecting the data a few times by pressing Clear Graph between runs until you have clean data that represents one steady pass.
- Use Zoom to Data to fit the graph around your measurements. When ready, click Capture Graph to copy the graph to your clipboard and paste it into your worksheet.
- Look back at your prediction drawing. Did you expect the V-shape to bottom out where it does, and does the sharp corner make sense given how the cart actually moved? Explain your reasoning in the worksheet.
- Enter your measured landmark position as H in the Desmos graph. This computes a new column, D, equal to $|S - H|$ for every data point, and plots D vs. time.
- Find the vertex of your D vs. time data - the point where distance is smallest. Record its time coordinate and its distance value.
- Turn on the absolute value model, $y = |x - h| + k$, and drag the h and k sliders until the V matches your data as closely as possible.
- Compare your fitted h to the time you'd expect the cart to reach the landmark, and compare your fitted k to how close you'd expect the cart to actually get on a discretely-sampled run. Explain any difference in the worksheet.
- After fitting the model, click Capture Graph and paste the graph image, along with your h and k values, into your worksheet.
- When finished, submit your worksheet according to your teacher’s directions.
- Try moving the landmark to a new position and repeating the roll. Does h change the way you'd expect? Does k stay about the same?
- Click Hide Directions to give yourself more space. Measure your landmark's position, click Zero Position at the start of the track, then click Start Collection and roll the cart in one steady pass past the landmark.