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How can we use the jetpack in these situations with consuming least energy?

1) Jumping to the top of k-block slope with gradient a°?
2) Jumping across the k-block gap?

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    When I looked at your profile, I was not surprised to find MathematicsSE at the top. What is your goal with this question? A code analysis? Experiments? Finding some info in wikis and combining it to a strategy? – Fabian Röling Nov 5 '18 at 12:12
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    @FabianRöling Some code analysis with mathematical modeling, I think. Hope it is not off-topic in Gaming.SE. – Kemono Chen Nov 5 '18 at 12:20
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    @KemonoChen I don't think it is off topic, but it might be a bit outside of the average scope of knowledge on this stack. Might take a while before someone who understands what you are asking, and knows the answer comes by. – Malco Nov 5 '18 at 14:40
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assuming fuel-powered jetpack; all calculations for k>>1

normal operation:

  • if jump is not pressed, do nothing
  • if 'forward' is pressed, accelerate by .02
  • increase upward motion by 0.2 until vertical speed is 0.6
  • consume factor 2

hover mode:

  • if 'forward' is pressed, accelerate horizontally by .02
  • increase upward motion by 0.2 until vertical speed is 0.6
  • if jump is pressed, slow positive ascend speed to .2; otherwise, negate any positive ascend speed
  • consume factor 1

basic minecraft movement algorithm:

  • apply horizontal acceleration in air: .02 (when trying to move)
  • add motion to position
  • add .08 downward motion
  • decrease downward motion by 2%
  • decrease horizontal motion by 9%

Going by intuition, the optimal normal mode jetpack usage to jump a gap seems to be a parabola with less than 45° initial angle (without any drag you'd want to spend all your delta-v right away; because of drag you need to reapply thrust now and then). The exact shape is quite difficult to calculate, since it doesn't follow physical laws too closely (vertical velocity cutoff; different drag per movement direction; tick-based algorithm).
Comparison "every third tick" normal mode to hover mode: normal mode costs 2/3 of hover, but hover mode reaches 1.43 times the horizontal terminal velocity due to accelerating permanently. Still, 1.43*0.66=0.96 -> burst firing is still more efficient than hover mode, even without using parabolic trajectories to improve usable velocity per delta-v.

Since conserving upward motion costs ~2 per 3 ticks in normal mode (comsume factor * gravity / thrust-acceleration), but 1 per tick in hover mode, you'll always want to be in normal mode when climbing slopes.
For optimal climb efficiency, accelerate to .6 upward speed, then fire every third tick to conserve upward velocity. When using this firing pattern, horizontal velocity will tend to be between around 0.29-0.3; this means the optimal height-to-depth ratio is ~2:1.

In both cases the efficiency should be better if accelerating quickly in the first phase, climbing until reaching the desired altitude, then firing a bit slower to loose upward impulse at a rate that makes you land exactly where you want to be.

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