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The Hyperloop, where passengers half-sit and half-lie in booths, allows distances that would take up to 10 hours by car to be covered in under an hour.
For example, travelling from Madrid to Algeciras (Cádiz province), the town on Spain's southernmost tip, would take just 42 minutes.
Musk's company Hyperloop One – a holding of the US firm Virgin - has contacted Spain's government to ask for public funding and to be allowed to set up a test and maintenance centre on the high-speed AVE rail link which connects the small inland town of Antequera (Málaga province) to Carmona (Sevilla province).
The AVE track, of 77 kilometres, will then be used to launch the first Hyperloop in Europe.
Spain's ministry of public works is currently in talks with Hyperloop One and may be willing to invest tens of millions as an initial deposit in the form of a loan.
Over the coming years, the Hyperloop system is expected to change the face of public transport with international as well as national links between major cities.
This could include a London-Madrid connection which would take just an hour and a half – faster than travelling by aeroplane.
And to get from Madrid to Morocco's northernmost city of Tangiers, a distance of 629 kilometres that would normally take seven hours of driving and at least another two on the ferry – not counting passport control and customs queues – would be as little as 47 minutes.
Madrid and Barcelona would only be 39 minutes apart, and in the longer-term future, Hyperloop One hopes to run a connection between Madrid and the Russian capital of Moscow which would take under four hours.
It could even lead to people commuting between European countries to work – and not just where they live close to a border.
A prototype of a first-class cabin on a Hyperloop designed to link up Dubai with Abu Dhabi in 15 minutes has been unveiled, although economy class cabin models have not yet been shown.
Typically, they would each have space for about 10 passengers, with 'in-flight' entertainment to make up for the lack of windows given that the Hyperloop is essentially a bullet train shooting through a giant tube suspended above motorways and railway lines at around 1,080 kilometres per hour.
Cargo cabins are also expected to be part of the system, since express deliveries are likely to bring in the bulk of the profits.
Photograph by Hyperloop One, showing a Hyperloop pod in a 'test tube'
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