Science and math textbooks love to use "placeholder" people. You know the ones. They’re always buying 47 watermelons or driving at a constant velocity for no apparent reason. But lately, a specific scenario has been popping up in physics and data literacy circles that actually matters for how we think. Angela and Carlos are asked to determine the relationship between two variables, and honestly, their struggle is basically a mirror of how we all mess up logic in real life.
Usually, this pops up in a physics lab context. They’re looking at a box on a ramp. They’re trying to figure out how the angle of that ramp changes the "normal force"—that’s just the fancy way of saying how hard the ramp pushes back against the box.
The Incline Problem Everyone Trips Over
Angela and Carlos aren't just names in a book; they represent two different ways of seeing data. In most versions of this problem, Angela is the one looking for a curve. She sees that as the ramp gets steeper, the box starts to feel "lighter" to the surface. She’s right, too. If you’ve ever tried to stand on a steep hill, you know your feet feel less "planted" than on flat ground.
But here’s where they get stuck: determining the actual mathematical relationship. Is it linear? Does it drop off in a straight line?
Nope.
The relationship between the normal force ($N$) and the angle ($\theta$) is actually $N = mg \cos \theta$. Because it involves a cosine, the graph is a curve, not a straight line. If Angela and Carlos just draw a straight line through their data points, they’re going to fail the lab. They have to "linearize" the data, which basically means they have to graph the force against the cosine of the angle to see that beautiful, perfect straight line that proves the law.
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Why This Matters Outside the Lab
You might be thinking, "Cool, I don't care about boxes on ramps." Fair enough. But the reason teachers keep using Angela and Carlos is to teach correlation versus causation.
We see this in the real world constantly.
Take the classic "ice cream and shark attacks" example. In the summer, ice cream sales go up. Shark attacks also go up. If Angela and Carlos were lazy scientists, they might "determine the relationship" as: Eating mint chocolate chip makes sharks hungry. Obviously, that’s ridiculous. The "lurking variable" is the heat. People buy ice cream when it's hot, and they also go swimming when it's hot. The two things are correlated, but one doesn't cause the other.
Angela and Carlos: The Friction Factor
Sometimes the problem adds a third person—usually Blake—to argue about friction. This is where it gets spicy.
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If the box is sitting still, it’s because static friction is holding it there. Angela might argue that the relationship is simple, but Carlos might point out that once the angle gets too steep, the relationship breaks because the box starts sliding.
In data science, we call this a "limit." Everything works until it doesn't.
- Linear relationships are rare in nature.
- Curves (like the cosine one Angela found) are everywhere.
- Variables almost never act alone.
When you're asked to determine the relationship between two things—like how much you sleep vs. how productive you are—it’s easy to be a "Carlos" and just assume it's a straight line. "If I sleep 10% more, I'll work 10% faster."
But it’s almost always a curve. At some point, sleeping more makes you groggier. The relationship "breaks."
How to Actually Solve It
If you’re staring at a homework assignment or a real-world data set and you need to determine the relationship like Angela and Carlos, do this:
- Plot the raw data first. Don't try to be smart. Just look at the dots.
- Look for the "bend." If the dots don't form a perfect spear, it's not a linear relationship.
- Identify the "Why." In the box-on-a-ramp problem, the "why" is gravity being split into two directions. In your life, the "why" is usually a hidden third factor.
- Test the extremes. What happens if the angle is 0? What if it's 90? If your "relationship" doesn't work at the edges, it’s not the right one.
Angela and Carlos are basically just a reminder that the world is rarely a straight line. We want it to be. It’s easier for our brains to handle. But whether you're looking at physics, the stock market, or why your plants keep dying, the real relationship is usually hidden behind a curve or a hidden variable you haven't noticed yet.
Next time you see a trend, ask yourself if you're seeing the whole ramp or just one small part of the slide.
To get this right, start by identifying your independent variable—the thing you're changing on purpose—and watch how the other one reacts without assuming a "straight-line" result. Collect at least five data points before you even try to draw a conclusion, and always look for that "third variable" that might be pulling the strings from the shadows.