Conservation of energy and the roller coaster

A roller coaster starting from rest at the top of a 20.0 m hill reaches 19.8 m/s at the bottom if friction is negligible, whatever its mass.

Trading height for speed

A roller coaster has no engine on the way down. A chain or a launch gives it energy once, at the start, and from then on the ride is a long exchange between potential and kinetic energy. On the way down a hill potential energy turns into kinetic energy and the cars speed up. On the way up the next hill kinetic energy turns back into potential energy and they slow down.

When friction is small enough to ignore, the sum stays the same: KE + PE at the start equals KE + PE anywhere later. This is conservation of mechanical energy. It lets you find a speed from a height without knowing anything about the shape of the track.

A worked example from OpenStax

A coaster starts from rest at the top of a 20.0 m hill. At the bottom, all the potential energy has become kinetic energy: m g h = ½ m v². The mass appears on both sides and cancels, leaving v = √(2 g h) = √(2 × 9.8 × 20.0) = 19.8 m/s, about 71 km/h. A full train and an empty one reach the same speed, just as a heavy and a light stone fall together when air resistance is small.

If the coaster passes the top already moving at 5.00 m/s, its starting kinetic energy adds on: ½ v² = ½ v₀² + g h gives v = √(5.00² + 2 × 9.8 × 20.0) = 20.4 m/s. That is faster than before, but only by 0.6 m/s, far less than the 5 m/s head start. Because energies add and speeds come from a square root, a head start helps less than you might guess.

Friction turns energy into heat

Real tracks and wheels have friction, and air pushes back. These forces do negative work, so the mechanical energy, KE + PE, slowly shrinks. The energy is not destroyed; it becomes thermal energy in the wheels, the rails and the air. The full law of conservation of energy keeps the books balanced: KE + PE + heat stays the same. In the lab, friction is a sliding coefficient μ, and the heat equals μ m g times the horizontal distance travelled.

Turn the friction up and watch the dashed line, the highest point the cart could still reach. Every metre of track lowers it. With μ = 0.05 the cart starting at 20 m clears the 14 m hill but cannot climb the 18 m one; it rolls back and forth until friction holds it in a valley. That is why a coaster's first hill is its tallest: each later hill must be lower than the energy left.

You can predict the lab's heat bar. With μ = 0.02 the heat produced is 0.02 × m × g for every metre of horizontal track, so after 130 m of track the cart has lost 0.02 × 130 = 2.6 m worth of height, whatever its mass. The friction slider in the lab is a model. Real coasters lose energy to wheel friction and air drag, and air drag grows with speed, so the loss is not the same on every metre. The bookkeeping, though, is exactly the same: what leaves the mechanical energy shows up as heat.

Efficiency

Machines face the same accounting. The useful output is always less than the energy put in, and the ratio is called efficiency: useful work out ÷ total energy in. OpenStax lists representative values. An electric motor reaches about 98 percent, a gasoline engine about 30 percent, a coal power plant about 42 percent and a solar cell about 10 percent. The human body manages about 20 percent when cycling or climbing but only about 2 percent swimming at the surface.