What if the second hill were taller than the first?

A roller coaster climbs a tall hill. Then whoosh! It zooms down and goes up the next hill.

The second hill is smaller. What if it were taller than the first one?

At the start of most roller coasters, a chain slowly pulls the cars up to the top of the first hill. After that, there is no motor at all: gravity does the rest, through dips, loops and more hills.

In this lab the first drop starts 40 meters up, the car goes around a loop, and then it meets a second hill 25 meters tall. What happens if we build that second hill 45 meters tall, higher than where the ride began?

A coaster released from rest at the top of its first hill carries a fixed budget of mechanical energy: gravitational potential energy mgH. Along the track that energy is traded between height and speed, and friction and air drag keep removing some.

This lab uses the roller-coaster model with a 40 m first drop, a 10 m-radius loop and lumped friction μ = 0.02, and raises the second hill from 25 m to 45 m. It also tests the frictionless case and asks how high the start must be to clear 45 m.

The car starts high up. It zooms down, goes around the loop and climbs the next hill.

That hill is smaller than the first one. Over the top it goes!

In the model, the car rolls down the 40-meter drop and reaches about 98 kilometers per hour at the bottom. It whirls around the loop, still going about 16.8 meters per second at the top.

Then it climbs the 25-meter second hill and gets over it easily, and the brakes stop it at the end. Along the way, friction from the wheels and the air turns about 30 percent of its energy into heat.

From 40 m with μ = 0.02 the model gives a bottom speed of 98 km/h, a loop-top speed of 16.8 m/s (the loop needs at least √(gr) = 9.9 m/s) and clears the 25 m hill, losing 30% of its energy to friction over the ride.

A frictionless car would reach √(2gH) = 28.0 m/s, about 101 km/h; the model shows 101 with μ = 0. The extra losses in the loop come from the model's friction law, which grows with how hard the track pushes on the car.

Now the second hill is taller than the first. The car zooms down and starts to climb.

It slows down, and down, and stops before the top. Then it rolls gently back.

Now make the second hill 45 meters tall, 5 meters higher than the start. The car goes down the drop and around the loop just as before. But as it climbs the tall hill it slows, stops before the top, and rolls back.

Is friction to blame? Switch friction off in the model and try again. The car still can't get over a 45-meter hill. With smooth wheels, the tallest second hill the car gets over is 29 meters; at 30 meters it rolls back.

With hill2 = 45 m the model reports "clears the second hill: no" for μ = 0.02 and also for μ = 0, since the car would need more energy than mgH at the crest. With μ = 0 it clears 39 m from a 40 m start (a hill exactly as tall as the start is a knife-edge case with no energy to spare, which the frictionless model counts as just clearing); with μ = 0.02 the highest hill cleared is 29 m (30 m fails).

To clear 45 m with μ = 0.02 the start must be raised to 63 m (62 m fails). Without friction, a 46 m start is enough.

Being high up stores energy. Going down turns it into speed.

Going up turns speed back into height. The car can't get more energy than it had at the start. So it can't go higher!

At the top of the first hill, the car's energy is stored in its height. Going down, that height energy turns into movement energy, so the car speeds up. Going up again, movement energy turns back into height energy, so it slows.

With no motor, the car can never have more energy than it started with. Friction also takes a little away all the time, turning it into heat. So no later hill can be taller than the start, and with friction every later hill has to be lower still. Our model treats the car as a dot on a smooth 2D track.

OpenStax's roller-coaster example shows that, with friction negligible, the loss of gravitational potential energy mgh becomes kinetic energy, the mass cancels, and the speed depends only on the starting speed and height, not on the path. Climbing to height h needs ½v² ≥ g(h − h_now), so a car starting from rest can never pass H.

OpenStax also explains that nonconservative forces such as friction remove mechanical energy as thermal energy that cannot be fully recovered. The model lumps rolling friction and drag into one μ times the normal push, uses cosine-shaped hills and a circular loop, and ignores train length and wind.

On a real roller coaster, a chain pulls the cars up the first hill.

That first hill is the tallest one. You can test it with marbles on a bendy track!

A lesson from Duke University's engineering program on TeachEngineering explains that on most coasters a chain pulls the cars to the top of the first hill, which is always the ride's highest point, because friction and drag start taking energy away at once.

The same lesson says that if a taller hill were placed in the middle of the ride, the cars could not get up it using gravity alone. It suggests testing this with marbles rolling along foam pipe insulation. The lesson adds that a few coasters do have engines that push the cars.

TeachEngineering's roller-coaster lesson (Duke University's Pratt School of Engineering) states that the first hill is always the highest point because friction and drag immediately begin robbing the car of energy, and that the car's total energy can never exceed what it had at the top of that hill. A taller middle hill would represent more potential energy, so a gravity-driven car could not climb it.

The lesson notes that while a few coasters do have engines pushing the cars, most rely on a chain lift and then gravity, and that cars can never regain their starting height unless a second chain lift is built into the track. This model has no motor and no second lift.