Mountain Gradients

pic of kids in scenic railway
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Cable trains are called “Funiculars”, and are used to go up and down very steep incline Gradients.

These gradients are too steep to have a train push or pull the carriages, and so the carriages are pulled up and down by steel cables.

The steepest Gradient cable train in the world is actually right here in Australia, at Scenic World in the Blue Mountains.

The train runs at an angle of 52 degrees, which is steeper than any normal train or road vehicle could ever go.

Here is a great two minute video about the Australian Cable train at the Blue Mountains.


Definition of Gradient

Gradient represents how steep a slope is in terms of its vertical change over a given horizontal Distance.

Mountain Gradients Definition PPT slide Copy
Image Copyright 2013 by Passy’s World of Mathematics


The Gradient is expressed as the Rise / Run

or as m = Vertical Change / Horizontal Change

A Gradient of m = 3/2 means that for every two meters we go horizontally, our hill goes up by 3 meters.

The Angle of Incline can be found by doing Inverse Tan-1 (3/2) which comes out as 56 degrees.

56 Degrees, or 3/2 is quite a Steep Gradient.


Fenicular Trains

stanserhorn carriage side on view
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Many Feniculars are powered by large diesel or electric motors, but the Stanserhorn Funicular in Switzerland is “self-powered” and requires no energy.

It is the perfect eco-friendly form of naturally powered transport.

How the Swiss train works under its own power is by having a continuous cable connecting a carriage going up with a carriage going down.

When the “down carriage” is at the bottom of the mountain, the “up carriage” is at its stopping point further up the mountain.

The force of Gravity on the train carriage going down the mountain, pulls the other train carriage that is going up the mountain.

Along the way there are passing sections.

These are located at a midpoint where it is exactly known the up train will meet the down train.

trains on stanserhorn
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Here is a photo of the two cable trains passing each other.

trains passing
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There are also “Funicular” self powered gradient cable trains in other parts of the world, such as the funicular which was built in 2005 along the Zambezi River, and is called the “Victoria Falls Eco-Lift”.

Mountain Gradients Victoria Falls Eco-Lift
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If you are interested in Feniculars, then check out the following website, which has lots of Feniculars featured from all around the world:


Mountain Cable Car Gondolas

cable car pic
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On the Stanserhorn Mountain, the Fenicular Train can only go so far up the Mountain.

At the train’s finishing point, the gradient of the mountain slope is too big for safe train travel.

Passengers get off the train, and board a cable car which goes to the top of the mountain.

By stringing the Gondolas of the Cable Car above the ground from towers of varying heights, the slope of the cables can be kept at an approximate 45 degree value where the cable cars can operate.

cable car side on view showing gradient
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Stanserhorn Videos

Here is a video which shows the Stanserhorn Fenicular journey.

It is a long eleven minute video which shows the whole trip.

We suggest watching the 5 mins to 5 mins 30 secs part where the two trains meet; and then watch from 9 minutes 20 seconds to 10 minutes 30 seconds part where a train comes into the mid mountain station to meet up with the cable car.


This second video shows the cable car ride section of the journey to the top of the mountain.

Be warned that it does have an interesting swiss folk song musical backing track.


Finally, if you are mega-fit, you can enter the race to the top of the Stanserhorn, which is shown in the following video.

The Race is conducted at the height of summer, when there is no snow on the mountain.


Finally, here is a good five minute video about why trains have trouble going up any type of hill.


Mountain Vegetation and Temperature Gradients

The air temperature is warm at the base of the mountain with the sun shining and blue sky, but very cold with ice and snow at the top of the mountain.

There is a decreasing gradient as we go up the mountain with temperature progressively getting lower.

The Vegetation and Soil also has a Gradient Change as we go up the mountain.

The vegetation and soil decreases follows along with the temperature decrease as we go up the mountain.

In the lower section where the train starts, there are houses, grasslands, flowers, and deciduous trees.

There is plenty of rich soil, and lush pastures for cows and other livestock.

pic of train and flowers
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However, as we ascend the mountain, the Gradient becomes steeper, there is less vegetation, with rocks amongst the grass, and only evergreen pine trees.

The vegetation change is due to the decrease in temperature, and the increase in rockiness, (less soil and more rock and gravel), as we go higher up the mountain.

Getting up onto mountain on train
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As we get right up near the top of the mountain, the sides are very steep, there is virtually no soil, and the only plants which can survive in the cold snow are pine trees.

Pine trees mountain
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As we get close to the top where it is very cold, only a few pine trees manage to cling to the steep sides of the mountain.

on way to top of stanserhorn
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The top of the mountain has a great view of the Swiss Alps, but the temperature is a lot colder than where the sun is shining on the lake at the base of the mountain.

Top of Stanserhorn Mountain
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Passy has been up the Stanserhorn Mountain in Switzerland, and it is a unique gradient experience which is highly recommended!

Mountain Gradient Math Summary
Image Copyright 2013 by Passy’s World of Mathematics

Ascending any big mountain, there are the gradients of the mountain slopes, the temperature gradient, and the soil and vegetation gradients.

Changes in these real world gradients make for a unique and scenic journey from the lakeside town of Luzern, up to one of the highest peaks in Switzerland.


Related Items

The Cartesian Plane
Plotting Graphs from Horizontal Values Tables
Plotting a Linear Graph using a Rule Equation
Plotting Graphs from T-Tables of Values
Finding Linear Rules
Distance Between Two Points
Real World Straight Line Graphs I
Real World Straight Line Graphs II



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