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Roller coaster and loops
Intermediate
Energy conservation gives the speed at any height on a track.
v = √(2g(h−y))Using energy conservation the speed at any height is v = √(2g(h₀ − h)) (frictionless). To pass the top of a loop of radius r, start from at least h = 2.5r.
Loop radius 10 m needs a start height of 25 m.
Starting height at the same level as the loop top isn't enough.
In more detail
At a loop, energy conservation gives the speed at each point, and a minimum speed at the top ensures the car stays on the track. The entry height must be high enough to supply that energy.
Real-world uses
- Roller coaster designers calculate the minimum drop height for each loop.
- Stunt performers in car loops rely on these energy calculations.
- A roller-coaster designer uses energy conservation to find the speed at each point on the track, so the cars arrive at the right speed.
Open the interactive lesson, with simulation and quiz →
Frequently asked questions
What height is needed?
Typically about 2.5 times the loop radius if starting from rest.
Does friction matter?
Yes, friction reduces the speed, so real rides need more height than the ideal calculation suggests.
How high must a coaster start to loop?
High enough that the speed at the top of the loop is adequate, which by energy conservation is a height above the top of the loop.
Related topics in Work, Energy & Power
Energy conservation in free fall · Energy on a pendulum · Energy in spring-mass systems · Energy loss to friction · General energy conservation · Power