Banked Curves – An Application to Bullet Trains

Banked Curves – An Application to Bullet Trains

I was recently in Japan for an amazing holiday (and am working pro bono for Tourism Japan, telling all of my friends that they HAVE to go), and one of the many wonderful experiences that I had was on a Japanese bullet train. These trains travel up to 320 km/h, and newer models that are being tested eclipse nearly 600 km/h. Going these speeds, I started to wonder how they don’t simply just go flying off the tracks as they turn corners. The sheer force of momentum as the train enters the corner (I would have thought) could rip the train right off the tracks. To put my hysteria in perspective, let’s assume that a train car weighs 43 tonnes (courtesy of wikipedia), and that the radius of curvature of the turn that the train is entering is 2500 m (courtesy of wikipedia), given this, the force that the train would exert on the tracks (as the tracks try to hold the car in place) would be:

Fc = mv2/r = 43,000 kg *

[(270 km/h) * (1000 m / km) * (1 h / 3600 s)]2 / 2500 m
Fc = 96,750 N

To put this force in perspective, this is greater than the force that would be applied to a person of a large African elephant were to sit on them!

With this in mind, I started doing research into banked curves, and how they are applied to bullet trains. This seemed to be the only solution, and I certainly trusted that the engineers in Japan had figured this out, but I still wanted to see how they went about solving this potential issue.

The first thing I discovered (to my absolute horror) was that not all bullet train engineers had figured this out, as there was in fact a major accident in Spain where a bullet train took a corner too quickly, causing an accident that killed 78 people. The next thing I found out (to my delight) was that banked curves have been used in the construction of Japanese bullet trains! Let’s see how this concept is applied in order to reduce the force that is acting on the tracks by the train as it makes a turn.


First, we should understand in simple terms how banked curves reduce the force exerted on train tracks. If we look at the two free body diagrams to the left, we can see that in the one on the left, as the train turns, the only force acting on the train is the force the tracks apply to the train in order to keep it traveling in a circle (centripetal force). Now, in the free body diagram on the right, we can see that there are 2 forces acting on the train: the force of the tracks, and gravity. Because the speed isn’t changing, the centripetal force is the same, but in a banked curve, gravity is helping to keep the train on the tracks, and reducing the amount of force that the tracks have to apply on the train to keep it traveling in a circle.

Let’s see this in action. Say we took the same train as above, but we sent it through a banked turn with a 10 degree bank. What force would the tracks apply to the train now in order to keep it on the track? Well, if the train stays on he track, that means that all of the forces acting on the train must cancel out. This means that:


Fc*cos(t) = Fg*sin(t) + Ftracks

(mv2/r)*cos(t) = m*g*sin(t) + Ftracks

[(43000 kg)([(270 km/h) * (1000 m / km) * (1 h / 3600 s)]2 / 2500 m]*cos(100) – 43000 kg * 9.81 m/s2 *sin(100) = Ftracks

Ftracks = 22,030.14 N

Look at how much banking the curve reduces the strain on  the train tracks. Banking the curve 10 degrees reduces the force on the tracks by 87%! Needless to say, I was very happy to hear that the curves were banked for Japanese bullet trains!