What’s the tallest loop-the-loop roller coaster that we could ever build and ride safely?

Amusement park physics taken to the next level.

Roller coasters have been the feature attractions of amusement parks for as long as they have been around. The thrill of being hurtled along a track of sharp dips and turns in a speeding car is unmatched by any everyday sensation. But one innovation in particular elevated the thrill of thrill rides to a completely different level — the vertical loop.

The first vertical loop, or “loop-the-loop” roller coaster was constructed in 1845 in Paris.

Vertical loop roller coaster in Paris, 1845

Hey, I know the image is pretty grainy, but I can see enough to know that it would take a lot to get me on that thing. Luckily for us, though, modern-day loop-the-loop coasters have come a long way in the past 170 years, using steel instead of wooden frames and changing the shape of the loop slightly to protect riders from potentially harmful accelerations.

And as they’ve become more safe, they’ve also become much bigger. Currently the title for the world’s tallest belongs to Six Flags’s “Full Throttle,” which features an immense loop of radius 25 meters!

“Full Throttle” roller coaster in Valencia, California

That’s pretty impressive, but what if we wanted to build the tallest possible loop-the-loop roller coaster that we could ever safely ride on? Sound like an impossible task? Read on and I’ll show you that with a bit of physics and a few simple calculations, this question is completely within our reach — and also a lot more interesting than you may think. But before we begin, let’s be scientific about this and define some things a little more precisely.

1. The Question

What is the radius of the tallest vertical loop roller coaster that we could ever build and ride safely on planet Earth? (It will soon become clear why the additional details are necessary.)

2. Implications

Alright, so now that we have a well-defined question, let’s unpack exactly what it implies.

First, the question asks for the “tallest” vertical loop. We can imagine that the taller the loop, the faster we have to be going to make it all the way around without falling off the tracks. So if we want to consider the tallest possible loop, then we must first consider the fastest possible speed. Could there be such a thing as a universal speed limit? In fact there is, and Einstein showed that nothing in the universe can travel faster than the speed of light. So let’s start there and see where it takes us (although, the speed of light is pretty damn fast, so we might run into some problems.)

Second, the question says “ride safely.” Sure this is all theoretical, but let’s try to be realistic. We don’t want any of our imaginary passengers fainting on our hypothetical roller coaster.

Third, the question specifies that this roller coaster is “on planet Earth.” It’ll become clear why this is important once we get into some actual roller coaster physics.

3. Assumptions

We’ll try to be as precise as possible, but answering questions like this almost always requires that we construct a simplified version of the real situation — a model, if you will. Plus, it’ll make our calculations a lot easier. The key, though, is not to make so many simplifications that the final result becomes worthless. So let’s go through a few important ones.

The shape of the loop is perfectly circular. If we were building one of those 19th century roller coasters, this wouldn’t be a problem. However, almost all modern-day vertical loops are not actually circles but instead “teardrop” shaped, better protecting riders from large accelerations. But it turns out that there are some simple, elegant, and most importantly readily available equations in physics that describe perfect circular motion. In order to use these in our calculations, let’s assume that our loop is a circle. There are no “energy losses” during the ascent of the roller coaster. This is a common assumption in mechanics problems and will allow us to ignore the heat energy transferred from the car to the tracks. In reality, no mechanical process is 100% efficient, but estimating the true efficiency of our coaster would be difficult and almost certainly unnecessary anyway. Our roller coaster is at sea level. Ok, fine. Truth be told, this one probably isn’t even worth including. But I’m mentioning it anyway to hint that we’ll need to know the distance between the center of the Earth and the bottom of the loop, which we will “approximate” to the radius of the Earth. Nonetheless, we could probably build our roller coaster on Mount Everest without affecting the final result too much.

With our question, approach, and assumptions now all in place, let’s get onto some physics, shall we?

4. Roller Coaster Physics

Note: This section is primarily for readers without a background in physics. If that’s not you, you might want to skip to the next part.

While physicists may not be the most artistically talented, they love drawing pictures. Remember, physics is grounded in reality. A well-drawn diagram can be just as useful a tool as equations and formulas. So let’s draw one.