What You Will Do:

• Connect two MotionCarts at the same time and collect position-time data from both at once, one graphed in red and one in blue.
• Model each cart's motion with a linear equation and treat the two equations as a system of linear equations.
• Activity A - start the carts on opposite ends of the track, move them toward each other at equal speeds, and predict: where will we meet?
• Activity B - give the slower cart a head start, send both carts the same direction, and predict: where will the faster cart pass the slower cart?
• Solve each system algebraically and graphically before collecting, then check your solution against the real intersection in Desmos.
• Complete the Google Docs worksheet and submit it according to your teacher's instructions.

Getting Ready:

• System of linear equations: two (or more) linear equations that use the same variables and are considered together. In this activity, each cart gets its own equation:
  $$S_1 = m_1 T + b_1 \qquad S_2 = m_2 T + b_2$$
  where $T$ is time (s), $S_1$ (red) and $S_2$ (blue) are the positions of Cart 1 and Cart 2 (cm), $b_1, b_2$ are their starting positions, and $m_1, m_2$ are their velocities (cm/s).
• Velocity and slope: velocity is speed with a direction. A cart moving toward larger numbers on the track has a positive slope. A cart moving back toward 0 has a negative slope, even if it is moving just as fast.
• Solution of a system: the pair $(T, S)$ that makes both equations true at the same time. On the graph it is the point where the red and blue lines intersect. For the carts, it is the moment and the place where the two carts are side by side.
• Solving algebraically: at the solution both positions are equal, so set the equations equal to each other and solve for $T$:
  $$m_1 T + b_1 = m_2 T + b_2$$
  Then substitute that $T$ into either equation to find the meeting position $S$.
• Types of systems: two lines with different slopes cross at exactly one solution. Two lines with the same slope and different intercepts are parallel and have no solution - the carts never meet. Two identical lines have infinitely many solutions - the carts move together the whole time.
• To review solving systems of linear equations by graphing before starting, click the video thumbnail below:
  
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• Click this link MotionCart - Systems of Linear Equations to open the worksheet in a new browser tab. Click Make a copy to save your version to your Google Drive.

Setting Up Your Track:

1. You will need masking tape, a ruler or meter stick, a marker, two MotionCarts, and three people: one driver for each cart and one timer.
2. Clear a flat, straight stretch of table or floor at least 130 cm long.
3. Make two parallel lanes about 10 cm apart so the carts can roll side by side and pass each other without colliding. Lane 1 is for Cart 1 (red) and Lane 2 is for Cart 2 (blue).
4. Use the ruler to place a strip of tape across both lanes at 0 cm, then every 10 cm up to 100 cm. Label each strip (0, 10, 20, ... 100).
5. Both carts must always face the same direction, with the USB-C port (the front of the cart) pointing toward the 100 mark. Never turn a cart around - a cart that needs to move toward 0 rolls backward. Always line up the front edge of a cart with a tape mark.
6. Pacing: the timer counts seconds out loud at a steady pace (a metronome app set to 60 beats per minute helps). To move at 10 cm/s, a driver moves the cart one tape mark every count. To move at 5 cm/s, the driver moves one tape mark every two counts, reaching the halfway point between marks on each count.

Using Your MotionCarts:

1. For detailed setup instructions and how to use your MotionCart, complete the Getting Started with MotionCart activity first.
2. Turn on the first MotionCart and click Connect. Select it from the browser's device list - it will become Cart 1 and graph in red.
3. Turn on the second MotionCart and click Connect again. Select it from the list - it will become Cart 2 and graph in blue.
4. Place both carts in their lanes with their front edges on the 0 mark. Click Zero Position and choose 1, then click Zero Position again and choose 2. The reading in the top bar should show 0 cm for both carts.
5. Roll each cart forward to the 20 mark and confirm its position reads about 20 cm. If a cart reads negative, click Change Sign, choose that cart, and check again.
6. Important: after both carts are zeroed at the 0 mark, do not zero them again. To give a cart a different starting position, roll it to its starting mark - the cart keeps counting, so it will read its true starting position. This is what gives each equation the correct intercept $b$.
7. Collection begins the instant Start Collection is clicked. The timer counts "3, 2, 1, Go" and clicks Start Collection on "Go" as both drivers begin moving. Collection stops automatically at the right edge of the graph, or when you click Stop Collection.

Activity A - Where Will We Meet?

1. The situation: Cart 1 starts at the 0 mark and moves forward at 10 cm/s. Cart 2 starts at the 100 mark and moves backward, toward Cart 1, at the same speed of 10 cm/s. Both start at the same moment.
2. In your worksheet, write an equation for each cart in the form $S = mT + b$. Think carefully about the sign of Cart 2's slope.
3. Solve the system algebraically to predict the time $T$ and position $S$ where the carts will meet.
4. Sketch your prediction below: choose Red Pen and draw Cart 1's line, then choose Blue Pen and draw Cart 2's line. Mark where you predict they cross. Click Erase Drawing to start over, and Capture Drawing to copy your sketch into your worksheet.
Red Pen (Cart 1) Blue Pen (Cart 2) Capture Drawing Erase Drawing

5. Collect the data: with both carts still zeroed at the 0 mark, roll Cart 2 forward to the 100 mark and check that it reads about 100 cm. Leave Cart 1 on the 0 mark. Click Clear Graph, then count "3, 2, 1, Go" and click Start Collection. Both drivers keep pace with the count - Cart 1 forward, Cart 2 backward - and keep going as the carts pass each other side by side.
6. Repeat with Clear Graph until the two lines look straight and steady. Remember to roll the carts back to their starting marks without zeroing them.
7. Find the intersection in your data: expand the data table in Desmos and look for the row where $S_1$ and $S_2$ are closest. Record that $T$ and position.
8. Check your algebra on the graph: in an empty Desmos expression line, type your Cart 1 equation using $y$ and $x$ (for example $y = 10x$), and your Cart 2 equation on the next line. Click where the two lines cross to show the intersection point. How close is it to your data and to your algebraic solution?
9. Why did the carts meet where they did? Explain what equal speeds have to do with the meeting position. Predict where they would meet if Cart 2 started at the 80 mark instead.
10. Click Capture Graph and paste your graph into your worksheet.

Activity B - Where Will the Faster Cart Pass?

1. The situation: Cart 2 gets a 30 cm head start and moves forward slowly at 5 cm/s. Cart 1 starts at the 0 mark and moves forward faster, at 10 cm/s. Both start at the same moment and move in the same direction.
2. In your worksheet, write an equation for each cart. Before solving, decide which line will start higher on the graph and which line will be steeper.
3. Solve the system algebraically to predict the time and position where the faster cart passes the slower cart.
4. Sketch your prediction below using the red and blue pens, and mark where you predict the lines cross. Capture your sketch into your worksheet.
Red Pen (Cart 1) Blue Pen (Cart 2) Capture Drawing Erase Drawing

5. Collect the data: return both carts to the 0 mark without zeroing (each should read about 0 cm), then roll Cart 2 forward to the 30 mark and check that it reads about 30 cm. Click Clear Graph, count "3, 2, 1, Go," and click Start Collection. Cart 1 moves one tape mark per count, Cart 2 one tape mark every two counts. Cart 1 passes Cart 2 in the next lane.
6. Repeat until you have a clean run, then find the intersection in your data table and with your typed equations, just as in Activity A. Compare both to your algebraic prediction.
7. Before the lines cross, which cart is ahead? After they cross? How can you tell from the graph alone?
8. The gap between the carts starts at 30 cm and shrinks by the difference in their speeds every second. Use this idea to explain your passing time, and show that it matches $T = \dfrac{b_2 - b_1}{m_1 - m_2}$.
9. Click Capture Graph and paste your graph into your worksheet.

Going Further:

• No solution - repeat Activity B, but move both carts at 10 cm/s. Predict what the graph will look like, then test it. What does it mean about the system when the lines are parallel, and what does it mean for the carts?
• Infinitely many solutions - start both carts at the 0 mark and move them together at the same speed. How many solutions does this system have, and why?
• Design a meeting - choose a target, such as "the carts meet at $T = 4$ s at the 60 mark." Pick starting positions and velocities that make it happen, write the system, test it, and report how close you came.

Let's Go:

• Click Hide Directions to give yourself more space. Work through Activity A and Activity B in order, capturing and pasting each graph into your worksheet before moving on.