What You Will Do:

• Sketch your prediction of what a graph of light intensity (I) vs. position (S) will look like as the MotionCart rolls through your phone's flashlight beam.
• Collect light intensity data in Desmos by rolling the MotionCart perpendicular across your phone's flashlight beam.
• Use two sliders to mark the edge and the center of the beam in your data, and use an inequality to find the width of the beam.
• Enter that width, along with the distance you measured from the phone to the cart's path, into a Desmos geometry construction that reads out the beam's angle - the main measurement you're after in this activity.
• Complete the Google Docs worksheet and submit it according to your teacher's instructions.

Getting Ready:

• Light intensity: a measure of how much light energy reaches a surface, measured in lux (lx).
• Position: how far the cart has moved from its zero position, measured in centimeters (cm). In this activity, position represents the cart's location as it rolls across the width of the flashlight beam.
• Flashlight beam: the cone-shaped region of light that spreads outward from the LED. Inside the beam, the light sensor picks up measurable brightness; outside its edges, the intensity is 0 (or very close to it). The beam's width - how wide that bright region is - grows the farther you measure from the LED.
• Inequality: a mathematical statement that compares two expressions using $<$, $>$, $\le$, or $\ge$ instead of an equals sign. Rather than describing one exact value, an inequality describes a whole range of values that make the statement true. Two inequalities can be combined into a compound inequality to describe values that fall between two bounds. For example, the compound inequality below describes every cart position S where the light sensor is inside the beam - between the beam's start, $S_0$, and its end, $S_0 + W$:
  $$S_0 \le S \le S_0 + W$$
  For any position S outside that interval, the cart is outside the beam and the light intensity is 0. This is the same idea you'll use later to describe the boundaries of your own data.
• To review how a two-variable inequality is graphed as a shaded region, like the one you'll see in the Desmos graph below, 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. As the MotionCart rolls in a straight line perpendicular to your phone's flashlight beam, it starts outside the beam (where the light sensor reads 0), crosses into the beam, passes through its brightest point, and continues out the other side. Based on that, sketch your prediction of what a graph of light intensity (I) vs. position (S) would 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.
Capture Drawing Erase Drawing

Collecting Data:

1. During collection, light intensity is recorded in the I column of the Desmos data table alongside position in the S column.
2. Dim the room lights so your phone's flashlight is the main light source reaching the sensor.
3. Prop your phone upright on the table with the flashlight LED near table top, facing straight across the table.
4. Measure the distance straight across from the phone's LED to where the MotionCart's light sensor will pass as it rolls by — a common setup is about 6 cm, measured to the sensor itself (roughly half the cart's width out from its edge). Record this distance; you'll need it later to find the beam angle.
   
5. Line the cart up so it will roll in a straight line perpendicular to the beam, and place it so its sensor starts outside the beam.
6. Click Connect Cart and pair your MotionCart. With the cart still at that starting spot, click Zero Light - this records whatever the sensor is currently reading (room light, screen glow, etc.) as a baseline and subtracts it from every reading after, so a Light value of 0 really means "no more light than when you started," not literal darkness.
7. Still at that same starting spot, click Zero Position so Position also reads 0.00 cm.
8. Click Start Collection, then roll the cart steadily in that straight line, continuing past the point of maximum brightness before stopping.
9. Collection will stop automatically when the cart is held still for a few seconds, or you can click Stop Collection to stop early.
10. Repeat collecting the data a few times by pressing Clear Graph between runs until you have clean data that represents the phenomenon.
11. 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.

Analyzing Your Data:

• Scroll down to the Desmos graph.
• Drag the N zero slider until the point it selects sits right at the edge of your data - the first point where the intensity rises above 0. This marks where the cart entered the beam.
• Drag the N final slider until the point it selects sits at the center of the beam - where the intensity reaches its maximum. Because your data only rises cleanly on one side before the sensor gets blocked, this is the point you'll mirror around.
• The shaded region that appears is built from those two points using an inequality; it's a check on your slider placement, not the actual shape of the beam - that comes later in the geometry below.
• Once N₀ and N_f mark two different points, the width (W) of the beam is calculated automatically as $W = 2\left(S_f - S_0\right)$ - doubling the distance from the edge to the center, since only that rising side of your data can be trusted. Record this value.
Capture Graph
• Scroll down to the Flashlight Construction geometry above. Drag the horizontal point labeled drag so its measurement matches the distance you measured earlier from the phone to the cart's path.
• Drag the vertical point labeled drag so its measurement matches half the beam width, W ÷ 2, that you just calculated.
• The angle shown at the flashlight updates automatically once both measurements are set - this is your measurement of the beam's angular width, the goal of this activity.
• Click Capture Graph above the geometry to copy it to your clipboard, and paste both it and your beam width and angle into your worksheet.
• When finished, submit your worksheet according to your teacher’s directions.

Going Further:

• Repeat the whole measurement again, but this time change the distance d between the phone and the cart's path before you re-measure. Compare the new angle to your first oneshould it stay about the same even though W came out different?
• If your phone lets you adjust the flashlight, try narrowing or widening the beam and repeat the measurement. Does the beam width W change, the beam angle, or both?

Let’s Go:

• Click Hide Directions to give yourself more space. With the phone propped up and the cart's starting spot outside the beam, connect your cart and click Zero Light, then Zero Position, then click Start Collection and roll the cart across the beam and past its maximum.