All the tech we rely on, from cars to smartphones, was engineered using physics. You don’t need to know the science to use these things. But a well-rounded human should understand at least some of the key concepts—along with some music, art, history, and economics. Robert Heinlein said it all in Time Enough for Love:
“A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects.”
So, in the interest of not being insects, here’s my top-five physics equations you should know.
1. Newton’s Second Law
I’m sure you’ve seen this one before—it’s over 300 years old, and it’s popular for science memes and T-shirts. It says the net force on an object equals its mass (m) times acceleration (a). But what does that really mean? It’s all about interactions—like when you kick a soccer ball or drop a water bottle on the floor.
Newton’s second law says we can describe these interactions with the concept of “force.” And what do forces do? The net force on an object changes the object’s motion. But wait! There’s a bunch more cool stuff in this simple-looking equation.
See those arrows over F and a? That indicates variables that are vectors, meaning they contain more than one piece of information. For example, if someone asks you to “socially distance” yourself by 1 meter, where would you end up? Who knows? You could go 1 meter to the east or west or 39 degrees from north. The distance by itself isn’t the full story; you also need to specify a direction. This is true for both the forces and the acceleration. Other quantities (like mass or temperature) don’t have direction. We call those scalar values.
Newton’s second law is super useful, but weirdly, people don’t seem to believe it. The common misconception is that a constant force makes an object move at a constant speed. What this equation says, rather, is that if you push on an object with a steady force, it will keep accelerating.
Why do people get this wrong? It’s because you almost never have just one force acting on an object. If you kept a steady pressure on your car’s gas pedal, and that was the only force on the car, believe me, you’d keep going faster and faster. But in reality, there’s wind drag pushing in the opposite direction, which partially offsets the force from your engine.
By using forces to describe interactions, we can do a bunch of awesome things—like model the motion of a basketball in the air, an accelerating drag racer, or even the motion of a binary star system.
2. The Wave Equation
If you take a long string and shake one end, you will create a disturbance. Guess what, with that simple shake you’ve made a wave pulse, which travels along the string. It’s pretty cool and very easy to do. I’m in Louisiana, so naturally I used a string of Mardi Gras beads.
Don’t let this one scare you—it’s a differential equation, which is calculus. But the idea is simple. If you picture a coordinate plane with a string laid out on the x-axis, it says the position of the string in the vertical (y) direction depends on both time (t) and the location of that part of the string (x).
Using this, we can model a pulse on a string to show that it moves with a velocity v. For actual real strings, this wave velocity depends on both the mass per length as well as the string tension. Here’s a fun example of a wave on a massive string in case you want to see some more math.
Why should you care? Well, it turns out that waves are all around us. Light is an electromagnetic wave. Same for the radiation in your microwave oven and your Wi-Fi signal. Sound is a pressure wave in the air. You can also pluck a string with both ends fixed, causing the wave to bounce back and forth—this is called a “standing wave,” and it’s how people play guitar. Kind of a big deal.
As you will see shortly, we can even describe the behavior of super tiny things like electrons with a wave-ish equation.
3. Maxwell’s Equations
Heh. I’m cheating a little by including four equations as just one, but it’s a package deal. Maxwell’s equations basically describe everything you need to know about the electric field (E) and the magnetic field (B) and how they’re related. The electric field is a way to describe interactions with electric charges (like electrons and protons), and the magnetic field describes what happens when those charges are moving (either in an atom or in an electric current).
Pretty much everything around you has something to do with the electric-magnetic interaction. When you push on a wall, why doesn’t your hand pass through it? After all, the wall is not a solid object, it’s made up of discrete atoms. It’s because of an electric interaction between the electrons in your hand and those in the wall, which repel each other. Then of course there are things like light bulbs, electric motors—oh, and computers.
Wait! There’s something even more important about Maxwell’s equations. You can use these to show that an oscillating electric field creates a changing magnetic field, and this changing magnetic field creates an electric field. Essentially, you can make a wave (with the wave equation) for electric and magnetic fields in the same way that a wave moves on a string. The speed of this electromagnetic wave (in a vacuum) would be 2.99 x 108 meters per second—which just happens to be the speed of light. So, light is indeed an electromagnetic wave. That’s kind of a big deal.
4. Schrödinger’s Equation
Schrodinger’s Equation
Schrödinger’s equation is the mathematical model at the heart of quantum mechanics. Although we can use Newton’s second law to understand the behavior of a baseball or the moon orbiting the Earth, this doesn’t work when we get to super small things like electrons and protons. It turns out that many of our ideas about motion just don’t work at the subatomic scale. Richard Feynman once said, “If you think you understand quantum mechanics, you don’t understand quantum mechanics.”
That said, let’s take a quick look at the Schrödinger equation. The version above is called the time-dependent Schrödinger equation in one dimension. See the variable Ψ (psi)? This is called the wave function. It’s a way to represent the probable location of a particle, since we can’t actually calculate the trajectory. We call it a wave function because it has a wavelike solution—which is nice, since we can then use our math techniques for dealing with waves.
Notice that this equation relates a time rate of change (on the left) and a space rate of change (on the right)—like the wave equation we explored earlier. It might seem weird that this includes an imaginary number (i is the square root of –1), but these often pop up in physics models—they’re quite useful to represent oscillations.
Let’s look at another awesome part of this equation: â„Ź. This is called the reduced Planck constant, and it gives a relationship between energy and frequency at the quantum level. (For kicks, you can measure this fundamental constant using different colored LEDs.)
Now, if you’re all in on quantum, and you want to tattoo Schrödinger’s equation on your arm, I’d suggest using this shorter version:
It’s basically the same thing. You still have the dependence on time on the left. The space part is replaced with the Hamiltonian operator (an H wearing a hat). Finally, |Ψ> is called a state vector. It’s just a different way to represent the wave function Ψ.
OK, but why should you care about the quantum realm? Well, even though you can’t go there like Ant-Man, we often deal with things at the atomic level. Just think about a single water molecule. It’s an interaction between one oxygen atom and two hydrogen atoms. Even this simple molecule is very complicated, but it can be modeled using Schrödinger’s equation. If you don’t like water (you should), there’s a whole range of technologies based on quantum mechanics: lasers, atomic clocks, LEDs, and of course semiconductors (used in the computer you’re reading this on).
5. Einstein’s Energy-Mass Equivalence
If you ask any random person to give you a physics equation, there’s a good chance you’ll get this one. In short, it shows a relationship between energy (E) and mass (m) with the constant c for the speed of light (2.99 x 108 meters per second) But wait! Although this version is what everybody knows, a more complete version looks like this:
This takes the particle’s velocity (v) into account, so we can get an expression for the total energy of a particle. If the velocity is much lower than the speed of light, then the energy is approximately:
That 1/2mv2 might look familiar. It’s the kinetic energy of an object. So, we can say that the energy of something is the sum of its “rest mass energy” (mc2) and its kinetic energy.
It’s crazy though. If you throw a baseball, it obviously has kinetic energy because it’s moving. But Einstein’s equation says it also has energy when it’s stationary. Let’s look at some actual values. Suppose a baseball (mass of 0.149 kilograms) is moving with a speed of 40 meters per second (a professional pitcher speed). The ball would have a kinetic energy of 119 joules, but the rest mass energy would 1.33 x 1018 joules. You can tell that’s big, right? But you have no idea.
In 2022 the US used 4.07 trillion kilowatt-hours of energy. If you convert that value, you get 1.46 x 1019 joules. So, if you were able to take 11 baseballs and convert all of their mass into electrical energy, it would be enough to run the US for a year.
This is exactly what happens in a nuclear power plant. Large-mass elements (like uranium) are hit with neutrons so that they break into pieces. However, the total mass of all the parts that it breaks into is less than the original uranium mass. The lost mass is converted to energy. Since the c term in the equation is squared, a little bit of mass gives a lot of energy.
Oh, maybe you don’t like nuclear power plants. Fine. Do you like other types of energy? Do you like to eat? Do you like weather? All of these things depend on this large object in the sky called the sun. Yes, the sun produces light from a nuclear reaction in its core. This light is the source of most other forms of energy production. It helps plants grow (you can eat those), and animals eat those plants (you can also eat the animals). This solar energy heats the surface of the Earth to produce changes in temperature and weather. The sun is kind of a big deal.
So, perhaps E = mc2 may be the most famous physics equation, but it’s also the most important. I do like to eat stuff.