Sometimes the universe is just too complicated to analyze.
Heck, if you take a tennis ball and toss it across the room, even that is practically too complicated. After it leaves your hand, the ball has a gravitational interaction with the Earth, which makes it accelerate toward the ground. The ball is spinning as it moves, which means that there could be more frictional drag on one side of the ball than on the other. The ball is also colliding with some of the oxygen and nitrogen molecules in the air—and some of these molecules end up interacting with even more air. The air itself isn't even constant—the density changes as the ball moves higher, and the air could be in motion. (We normally call that wind.) And once the ball hits the ground, even the floor isn't perfectly flat. Yes, it looks flat, but it's on the surface of a spherical planet.
But all is not lost. We can still model this tossed tennis ball. All we need are some idealizations. These are simplifying approximations that turn an impossible problem into a solvable problem.
In the case of the tennis ball, we can assume that all the mass is concentrated at a single point (in other words, that the ball has no actual dimensions) and that the only force acting on it is the constant downward-pulling gravitational force. Why is it OK to ignore all those other interactions? It’s because they just don’t make a significant (or even measurable) difference.
Is this even legal in the court of physics? Well, science is all about the process of building models, including the equation for the trajectory of a tennis ball. At the end of the day, if the experimental observations (where the ball lands) agrees with the model (the prediction of where it will land), then we are good to go. For the tennis ball idealization, everything works very well. In fact, the physics of a tossed ball becomes a test question in an introductory physics class. Other idealizations are harder, like trying to determine the curvature of the Earth just by looking at this super-long terminal in the Atlanta airport. But physicists do this kind of thing all the time.
Perhaps the most famous idealization was done by Galileo Galilei during his study of the nature of motion. He was trying to figure out what would happen to a moving object if you don't exert a force on it. At the time, just about everyone followed the teachings of Aristotle, who said that if you don't exert a force on a moving object, it will stop and remain at rest. (Even though his work was around 1,800 years old, people thought Aristotle was too cool to be wrong.)
But Galileo didn’t agree. He thought it would keep moving at a constant speed.
If you want to study an object in motion, you need to measure both position and time so that you can calculate its velocity, or its change in position divided by the change in time. But there is a problem. How do you accurately measure the time for objects moving at high speeds over short distances? If you drop something even from a relatively small height, like 10 meters, it takes fewer than 2 seconds for it to reach the ground. And back around the year 1600, when Galileo was alive, that was a pretty difficult time interval to measure. So, instead, Galileo looked at a ball rolling down a track.
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Now for the idealization: If a ball starts rolling down a completely horizontal track, it will slow down a little bit as it moves. But if you create a track so that it's just slightly inclined above the horizontal, it's not too difficult to show that the ball increases speed during its motion. And if you get the track at just the right angle, you can push the ball and it will move along at a constant velocity—it doesn't speed up or slow down. Galileo used this to argue that if you could absolutely remove all the friction between the ball and track, so that no forces would act on the ball, it would move at constant speed—and Aristotle would be wrong.
Just to be clear, Galileo never devised an experiment with a ball that actually had zero forces acting on it. He just made an idealized version.
Would it even be possible to have a ball that doesn’t have any forces acting on it? It’s possible, but it would be very difficult. First, you would have to remove the air, so that there is no air drag force on the ball. Second, the ball would have to move without touching anything. And third, you’d have to remove the gravitational force. Yes, you could put it out in deep space, away from any massive objects. However, even a far-away star will exert a gravitational force on an object. Even nearby humans looking at this moving ball would exert a gravitational force. (It would be small, but it would be there.) So in the end, you would likely still need to make an idealization.
How about another example? Suppose you want to calculate the gravitational interaction between two people standing 1 meter apart.
We have the following model for the gravitational interaction between two objects:
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In this expression, G is the universal gravitational constant and r is the distance between the two objects with masses m1 and m2. But there's a problem. This model assumes that the two masses are just points without any dimensions. Clearly people are not just points.
So let's just make the idealization of a spherical human with all the mass concentrated at the center of mass. Then we can use the gravitational formula above to calculate the force. Yes, it's technically wrong—but if your goal is to show that the gravitational force is tiny (and it is) then it doesn't actually matter if you have real humans or point humans.
(You can use the same idealization when you calculate the gravitational force between a human and a billiard ball, which I did here.)
Let’s try another: an idealization with light. Suppose I get a red laser pointer and shine it onto a thin film of oil to create an interference pattern. In physics, we like to pretend that the laser light is collimated and monochromatic. Collimated light consists of electromagnetic waves all traveling in the exact same direction. Lasers make a very tight beam of light that is mostly collimated, but not exactly. Monochromatic means that the light is one single wavelength. Again, a red laser is mostly just one wavelength, but not exactly.
However, when we do analysis with the red laser, we can make the idealized approximation that the light is indeed collimated and monochromatic. We can actually shine a laser on thin films and measure the interference pattern. As with all physics, if the theoretical calculation agrees with experimental data—it’s a win.
Of course, sometimes an idealization just doesn’t work. Imagine trying to calculate the curving motion of a soccer ball after it’s kicked. If you assume it’s a point mass that doesn’t rotate and doesn’t interact with the air, it just won’t work. In this case, the rotation and drag effects might be small, but they are crucial to figuring out where the ball will go.
The real world is messy. But sometimes when we can't handle the mess, we just make it simpler—and it works well enough to help us build a scientific model. Idealizations are like the Bitmoji of science. They don't show everything, but they show enough that we can figure out what's going on.
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