# The Energy Cost of a Roller Coaster Ride

## Introduction

 Fig. 1: Chain lift hill of Goliath at Six Flags Great America in Gurnee, Illinois. (Source: Wikimedia Commons)

Going to a roller coaster park can be expensive - on the order of \$100 for a single day admission. [1] An interesting question is then to ask, how much does it actually cost to ride a roller coaster? Or more specifically, what is the energy cost per ride? Does that account for a large portion of the admission ticket price?

## Energy Per Person

We can get a rough idea of the energy cost by doing some simple calculations. Let's take the roller coaster Goliath of Six Flags Great America (Gurnee, Illinois), a wooden rollercoaster famous for its height, speed, and angle of descent, as an example. [2] Goliath has one chain lift hill that takes the roller coaster train and its passengers up to a height of 180 feet (55 m). We can find the potential energy U at the top of the hill by using the formula

 U = m g h

where m is the mass, g is acceleration due to gravity (9.8 meters/second2), and h is 55 m. Since we are interested in the potential energy per passenger, we will set m to the mass of a typical person, 70 kg, plus the weight of the roller coaster train divided by the number of passengers. As a rough estimate for the train weight per person, we will use twice the mass of the person (140 kg). Carrying out the potential energy calculation, we obtain:

 U = (70 + 2×70) kg × 9.8 m/s2 × 55 m = 1.1 × 105 Joules

Since the roller coaster train comes to rest at the end of the ride due to friction and drag (the track, brakes, air, etc.), all of this energy is depleted; in other words, this represents the total energy cost per person per ride. We will ignore the energy required to run the operator controls, brakes, and other safety mechanisms by assuming it is much smaller than the energy required for the chain lift hill.

## Cost of Energy

The price of electricity in Illinois is about 12 cents per kilowatt-hour, or [3]

 \$0.12 kWh-1 / (3.6 × 106 J/kWh) = \$3.3 × 10-8 per Joule

If Goliath's chain lift and motor system operates at 100% efficiency, it thus costs

\$3.3 × 10-8 Joule-1 × 1.1 × 105 Joules = \$3.8 × 10-3

or 0.4 cents per person per ride. However, understanding that the lift hill system is not 100% efficient due to friction, etc., we can redo this calculation with a different efficiency, say 5% efficiency. This yields \$3.8 × 10-3 / 0.05 = \$0.075 or 7.5 cents per ride. (More digits than shown were used in some calculations.)

## Conclusion

Even when operating at 5% efficiency, 7.5 cents per ride on Goliath is miniscule compared to a \$100 admission ticket. This suggests that the cost of energy is likely not the driving factor in price; perhaps maintenance and inspections, initial design and construction, land cost, or other factors truly drive the cost of a day at a roller coaster park.

© Jonathan Timcheck. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

## References

[1] H. Martin, "Ticket Prices for Universal Studios, Disneyland and Other Theme Parks Are Rising Fast," Los Angeles Times, 6 Mar 2017

[2] D. Moran, "Six Flags Great America's Goliath Earns 3 Guinness World Record Titles," Chicago Tribune, 3 May 15.

[3] J. Jiang, "The Price Of Electricity In Your State," National Public Radio, 28 Oct 11.