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I thought of building a sort of subway but I was kind of bored of the regular walking, minecarts, and boats on ice. So I remembered flying machines. Is there a two-way design that is actually faster than going by the ordinary minecart? If so please give me a link or picture or at least some form of instructions.

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A minecart's top speed is 8 blocks per second (along one axis) .

A piston powered flying machine would be bound to redstone ticks, there are 10 redstone ticks per second. In theory, if you manage to push the flying machine every single redstone tick, then you could go at up to 10 blocks per second.

This would require an extended piston to instantly be replaced, or pushed forward. You would need at least a second redstone tick, just to move the piston back into place, making the theoretical top speed of a flying machine 5 blocks per second1, which is slower than a minecart.

1: 5 blocks average speed, it is however possible to go at up to 10 blocks per second in quick bursts.

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    This flying machine is the fastest I could find after looking it up for a bit, it's stated that it's impossible to stop it without breaking it completely, so it's definitely not two-way and it goes at 4.615 blocks per second, or a bit faster than half the speed of a minecart. – user232393 Sep 15 at 15:34
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    While the basic conclusion ("no") is correct, the answer is based on a flawed assumption that the machine must move at most 1 block per tick. There are machines where a part of it moves multiple blocks within the one tick; arranging multiple pistons to be activated within the same tick in a specific order you can achieve a burst of speed covering multiple blocks in the same tick. (melon&pumpkin harvester machines utilize this to push down every other block.) Unfortunately the complexity and required part count to achieve continuous high speed motion that way makes it unfeasible. – SF. Sep 16 at 12:22
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    @SF Do you have a reference for a relatively simple setup that moves a block, where the block is moved multiple blocks along a single axis within one tick (it doesn't have to push the block repeatedly and the setup doesn't have to reset itself)? If not, then it would be nice if you could link to one of those melon/pumpkin harvesters. I will look into it and update my answer accordingly. – user232393 Sep 16 at 12:39
  • Dissecting Minecraft pt 9, around 23:00. Later on, the update order mechanics is discussed. – SF. Sep 16 at 14:14
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    @SF The blocks in the example move at a speed of 1 block per tick. An observer has a 1 tick delay. What actually happens is this: in the first tick, one of the two pistons extends, the observer didn't fire yet, because of the delay. In the second tick the second piston extends, and then the observer ticks, as the delay is over at that point, causing the downwards piston to extend after "skipping" one block. There is no part that moves 2 blocks in one tick. If you try to extend this to move three blocks, then the downward facing piston would still extend after two blocks. – user232393 Sep 16 at 14:40

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