AP® Physics 1: Algebra-Based · Unit 3 · Topic 3.4
Roller Coasters, Loops, and Minimum Speeds
A coaster needs enough starting height to supply both the speed required at the loop's top and the energy lost to climbing there, which means releasing it well above the loop's peak, not just level with it.
At the top of a loop, gravity is the only thing available to pull the cart toward the center of the circle, since the track can only push, never pull a cart inward along its inside surface. That means gravity alone must supply the centripetal force, which pins down an exact minimum speed at the top: mg = mv2/r, so v_top = √gr. Any slower and the track would need to pull outward on the cart to keep it on the circular path, which it cannot do, so the cart leaves the track before reaching the top.
To find the release height, combine that speed requirement with energy conservation. The cart needs kinetic energy (1/2)mv_top2 = (1/2)mgr at the top, plus the potential energy mg(2r) it took to climb to that height. Setting the release height h so that mgh equals the sum of those two terms gives mgh = (1/2)mgr + mg(2r), so h = 2.5r, or 1.25 times the diameter. Notice the masses cancel, so the minimum height does not depend on how heavy the cart is.
This topic is where circular motion and energy conservation stop being separate tools and start working together: energy conservation tracks how speed changes around the loop, while the circular motion condition sets the one non-negotiable speed checkpoint at the top. The same combination shows up anywhere a path curves vertically, like the top of a hill on a road.
Key terms
- Centripetal force
- The net inward force needed to keep an object moving along a circular path.
- Minimum speed at the top
- The slowest speed, v = √gr, at which gravity alone can supply the centripetal force at the loop's peak.
- Energy conservation
- The principle that total mechanical energy stays constant when only gravity and normal forces do work.
- Release height
- The starting height above the bottom of the loop needed to give the cart enough energy to complete it.
Watch it explained
- How it worksHow do energy conservation and the circular motion condition combine to set the release height?
- Worked exampleFind the minimum release height for a cart to complete a frictionless 4 m diameter loop.
- Common mistakeWhy isn't reaching the top of the loop with zero speed good enough?
- Going furtherWhat changes about the required height if the loop is a teardrop instead of a circle?
Check yourself
End of topic.