Rollercoasters 2: Vertical G-force, hills and drops

In my last post about rollercoaster physics, I talked about what happens when a rollercoaster goes round banked and flat corners. This time, I’m going to take a look at what happens when the coaster train goes down drops and over hills – basically, every time your stomach gets left behind on a coaster!

First, let’s look at drops. Here’s a picture of a nice, gentle rollercoaster drop…

A massive, steep wooden rollercoaster drop.

The first drop on the El Toro rollercoaster, which reaches a massive 76 degree slope. Image: Dusso Janladde.

OK, so I lied about the nice, gentle part. As anyone who’s been on a rollercoaster knows, the steeper the drop, the faster the acceleration, and the longer the drop, the more the speed can build up (so this has gotta be pretty fast!). It’s all about the change in energy, and how fast it happens. Here’s another of my exciting diagrams:

Diagram of a drop on a rollercoaster, showing the weight, height and potential energy at the top.

At the top of a rollercoaster drop, the car - and the people inside it - have potential energy (PE).

To calculate the energy at the top of the drop, we use the formula for (gravitational) potential energy:

  • potential energy = mass x gravity x height of drop = weight x height of drop
The potential energy here is basically the energy stored at the top of the drop: the energy you potentially could gain. So, for example, on a skydive your potential energy at the top is the same as the kinetic energy (energy from the movement of you falling) at the bottom, because that’s the maximum energy you can gain from gravity.
In this case, the maximum energy that the car and its passengers can gain is restricted by the height of the drop: they can only drop a certain distance, so their maximum speed (and maximum energy) is restricted.
Once the rollercoaster train reaches the bottom of the drop, it’s reached its maximum speed:
Diagram of a rollercoaster drop, showing the kinetic and potential energies.

At the bottom, the kinetic energy (KE) is the energy that the rollercoasters have from speeding away.

We can work out the kinetic energy at the bottom in two ways: firstly, through the equation
  • kinetic energy = 1/2 x mass x velocity x velocity,
which only works if we know the speed. However, through the law of conservation of energy, we know that energy can’t be created or destroyed: so the energy at the top has to be the same as at the bottom. In other words, the potential energy at the top and the kinetic energy at the bottom are the same. Perhaps that’s not surprising, given that the potential energy is the maximum energy the rollercoaster can have at the bottom of the drop!
But what about hills, and that feeling of leaving behind your stomach you get when you go over them? Here’s a picture of the hill on Stealth at Thorpe Park (a truly amazing ride):
The massive hill on the Stealth rollercoaster, backlit by the sun.

The massive hill on the Stealth rollercoaster at Thorpe Park. Image: supernova3688

In this case, the rollercoaster is launched at a ridiculous speed from standing, shoots up one side of the hill before dropping down the other. The cars travel over the hill, held onto the track, so their directions of movement look something like this:
Diagram of rollercoaster cars going over the hill, showing the direction of movement of each car.

The cars are held onto the track, so their directions of travel are restricted.

But, it’s a different story for the passengers. Although (hopefully) they’re held in by harnesses or lap bars, they’re not connected to the track, so their directions of movement look like this:
Diagram showing the passenger directions of movement on a rollercoaster hill.

The passengers in a rollercoaster at the top of the hill keep on going up at the top before being pulled back down again.

It’s that bit at the top – where the passengers fly off before being pulled back by the harness – which gives that weightless, ‘air time’ experience that can often leave stomachs at the top.
I would suggest that you think about this next time you’re on a rollercoaster, and see if you can notice that air time experience occurring on the hills –  but I also know from experience that it’s pretty much impossible to think about anything on a coaster, so I’ll just leave you to think about it!
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