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geordief

The most energy efficient path in space-time from London to Glasgow

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I am not quite sure this should be in Relativity and I am not quite sure why I am asking the question .I am "fishing" a bit ,perhaps to flesh out my ideas but this is the question.

 

Suppose we are in London and need to be in Glasgow in a week's time , what is the most energy efficient method of getting there?

 

(obviously the method of transport and all other environmental factors are the same no matter which route is taken)

 

So ,the first option could be to leave directly ,arrive at Glasgow in 6 hours and remain in Glasgow for 6 days and 18 hours.

 

As an alternative itinerary we could also leave at once , go to Norwich in three days (traveling very slowly) ,remain there for 3 days and then take 24 hours to complete the journey to London.

 

A third possible itinerary might be to stay in London for 6 days and 12 hours and then journey to Glasgow taking 12 hours to do so.

 

In all cases we are going to travel approximately 500 miles due North and take 7 days overall to do so.

 

Out of all the infinite possible permutations involving velocities ,duration of velocities and lack of velocity is there one path between London and Manchester that is the most energy efficient when we know the direction from one to another and the time that is to be allowed for the journey ?

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The only variable you've identified here is the speed, which is tied in with kinetic energy. If all environmental factors are the same, the inefficiencies like friction don't matter. (except they will, because friction is path-dependent)

 

KE is equal to the work done to getting an object moving at speed v. To minimize the work, you want to move as slowly as possible, with no stopping and starting. 500 miles in 7 days, so you want to move a little more than 70 miles per day, continuously.

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The only variable you've identified here is the speed, which is tied in with kinetic energy. If all environmental factors are the same, the inefficiencies like friction don't matter. (except they will, because friction is path-dependent)

 

KE is equal to the work done to getting an object moving at speed v. To minimize the work, you want to move as slowly as possible, with no stopping and starting. 500 miles in 7 days, so you want to move a little more than 70 miles per day, continuously.

Thanks (I realized after I posted ) that any spacial detours were irrelevant.

 

Is it also irrelevant if speeds are allowed to be relativistic?

 

So my question is entirely trivial?

 

EDIT: Does the acceleration involved make the question any less trivial? Does it (the acceleration and deceleration )need to be spread out as much as possible over the entire length of the journey?

Edited by geordief

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Thanks (I realized after I posted ) that any spacial detours were irrelevant.

 

Is it also irrelevant if speeds are allowed to be relativistic?

 

So my question is entirely trivial?

 

EDIT: Does the acceleration involved make the question any less trivial? Does it (the acceleration and deceleration )need to be spread out as much as possible over the entire length of the journey?

 

 

The work-energy theorem, which shows that W = KE, does not depend on the details of the acceleration. From a physics standpoint, there is no optimal acceleration, or sub-optimal acceleration.

https://en.wikipedia.org/wiki/Work_(physics)#General_derivation_of_the_work.E2.80.93energy_theorem_for_a_particle

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The most energy efficient mode of travel is the gravity train 42 minutes from anywhere on Earth. But if you cannot build a friction free tunnel though the mantle a vactrain along the surface is just as efficient and faster if you have a perfect vacuum in your tunnel and efficient superconducting maglev track. All the energy put into the train to accelerate it, is recovered when you decelerate no matter how fast you go. In more realistic scenario's the slower the method of travel the more efficient it is as friction and air resistance increase exponentially with speed. But this can be offset by other things, when you divide a modern jetliners fuels consumption by the maximum number of passengers it has over 100 miles per gallon better than most cars at single occupancy.

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The most energy efficient mode of travel is the gravity train 42 minutes from anywhere on Earth. But if you cannot build a friction free tunnel though the mantle a vactrain along the surface is just as efficient and faster if you have a perfect vacuum in your tunnel and efficient superconducting maglev track. All the energy put into the train to accelerate it, is recovered when you decelerate no matter how fast you go. In more realistic scenario's the slower the method of travel the more efficient it is as friction and air resistance increase exponentially with speed. But this can be offset by other things, when you divide a modern jetliners fuels consumption by the maximum number of passengers it has over 100 miles per gallon better than most cars at single occupancy.

Am I right to think that this trajectory (through the Earth) is along a local geodesic in space time ? The initial acceleration will determine where (on the surface of the globe) you will "resurface" )?

 

Adding a time constraint to the trajectory adds nothing to the exercise ,does it? It would be impossible to "resurface" in Glasgow 7 days later by taking this route ,would it?

 

How long would it take for a journey from London to the far side of the globe and back up to London?If you were aiming for Glasgow the time would be determined at the outset by the length of the round trip (about 16,000 miles ) and the average speed.

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Am I right to think that this trajectory (through the Earth) is along a local geodesic in space time ? The initial acceleration will determine where (on the surface of the globe) you will "resurface" )?

 

Adding a time constraint to the trajectory adds nothing to the exercise ,does it? It would be impossible to "resurface" in Glasgow 7 days later by taking this route ,would it?

 

How long would it take for a journey from London to the far side of the globe and back up to London?If you were aiming for Glasgow the time would be determined at the outset by the length of the round trip (about 16,000 miles ) and the average speed.

 

42mins from anywhere to anywhere via gravity tunnel

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Bit of a surprise :)

 

There is a lovely proof - it was first shown by Newton or by Hooke depending on who you believe*

 

http://hyperphysics.phy-astr.gsu.edu/Hbase/mechanics/earthole.html

 

* must be a bugger to be a staggeringly great scientist like Hooke only to be in the same city at the same time as Newton - who was not only greater but also notoriously did not get along well with others

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So ,if there is any massive body with a surface equidistant from the centre of gravity any other moving body will (ignoring friction and if its velocity allows) intersect this surface in any two places after the same time interval?

 

It does not matter whether the initial trajectory is towards or from the centre of gravity?

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So ,if there is any massive body with a surface equidistant from the centre of gravity any other moving body will (ignoring friction and if its velocity allows) intersect this surface in any two places after the same time interval?

 

It does not matter whether the initial trajectory is towards or from the centre of gravity?

 

It is only for falling - ie only under gravity. That is no initial velocity (trajectory implies a velocity) - but the actual direction through the earth makes no difference; any straight tunnel will take 42 mins to get to the other end of under gravity alone

 

By the way; it wouldn't work :) You have huge tangential velocity which would mean that you would thump into the side/roof of the tunnel and bounce back and forth all the way through; the angular velocity remains the same all the way through a sphere - but the tangential velocity shrinks as you go deeper (ie the hole has lower tangential velocity but the vehicle retains its tangential velocity). It is actually even more complex than this - but enough to mean that it is pretty much non-viable; you have to be freefalling in a vacuum to cope with the speeds involved BUT you have to be constrained to stop the tangential movement smashing your vehicle to pieces. By the time we can do both - we will be using such technology that the need for gravity train will be past

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So ,if there is any massive body with a surface equidistant from the centre of gravity any other moving body will (ignoring friction and if its velocity allows) intersect this surface in any two places after the same time interval?

 

It does not matter whether the initial trajectory is towards or from the centre of gravity?

A gravity train works under the assumption that the train has zero velocity with respect to the surface at the start and that its path is restricted to a straight line between start and destination. It is these conditions which lead to the constant 42 min travel time.

 

If, on the other hand, you are considering a object following a purely free-fall path through the massive body, then its initial trajectory and speed upon intersection with the surface will have an effect on the time spent within the volume of the massive body.

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