Physics Question

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stardek
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Physics Question

Postby stardek » Sat Oct 31, 2009 9:31 pm UTC

Heres a physics question I got in class a while ago. The only way I know how to do it is trial and error so I figured it would be easier to ask here.

A rocket is hovering at 1000m and needs to reach the ground. It can't hit the ground at a speed faster than 5m/s. Gravity is 10m/s2 (I realize this is not very accurate but it does make the math easier :)) and there are boosters that accelerate the ship at 10m/s2. The boosters use 1 unit of fuel for every 1 second they are on. They can fire for 5 seconds in a row before taking a 1 second break. You have 19 units of fuel.

I would appreciate solutions for shortest amount of time and least amount of fuel used. An explanation of how the answer was found would be nice too :P If this needs clarification please say so, I am generally terrible at explaining stuff.
Last edited by stardek on Sun Nov 01, 2009 9:15 am UTC, edited 1 time in total.

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Re: Physics Question

Postby skeptical scientist » Sun Nov 01, 2009 12:31 am UTC

When you say the boosters accelerate the ship at 10 m/s2, do you mean in addition to countering the downwards force of gravity? Because otherwise the problem seems impossible.
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{delta}
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Re: Physics Question

Postby {delta} » Sun Nov 01, 2009 2:26 am UTC

In addition to that question, is the 5 second delay ONLY if they fire continuously for 5 seconds, or is it any time it accumulates 5 seconds of firing time there need to be a one second, no fire, delay?

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Anubis
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Re: Physics Question

Postby Anubis » Sun Nov 01, 2009 3:44 am UTC

skeptical scientist wrote:When you say the boosters accelerate the ship at 10 m/s2, do you mean in addition to countering the downwards force of gravity? Because otherwise the problem seems impossible.


Merely countering the downward force of gravity would mean there is no net force and hence no net acceleration. The question is worded poorly but I think we can safely assume that it is referring to a net upward acceleration of 10 m/s2.

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Re: Physics Question

Postby rflrob » Sun Nov 01, 2009 8:19 am UTC

If nothing else, for the fastest time to descent, it seems like you'd want to be firing your boosters as late as possible.

For fuel use, it seems like there ought to be a fairly broad subset of solutions that would minimize it. That's just energy conservation:
[math]mgh + fuel_{t = 0} = fuel_{t = landing} + 1/2 m (5m/s)^2[/math]
where you've converted your 19 units of fuel to some sort of equivalent energy with something like [imath]1 unit = 1/2 m \cdot (1 s \cdot 10m/s^2)^2[/imath]. That may not actually be correct... probably the thing to do is figure out how what it would accelerate you to in the absence of other forces.


Short of taking the limit of firing your boosters for infinitesimal periods of time, reducing your acceleration to something less than 10m/s^2, this seems like the kind of problem that won't necessarily have an nice, analytic solution. Discontinuities tend not to lend themselves to pretty formulas.
Ten is approximately infinity (It's very large)
Ten is approximately zero (It's very small)

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stardek
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Re: Physics Question

Postby stardek » Sun Nov 01, 2009 9:17 am UTC

{delta} wrote:In addition to that question, is the 5 second delay ONLY if they fire continuously for 5 seconds, or is it any time it accumulates 5 seconds of firing time there need to be a one second, no fire, delay?


The delay is only after firing for 5 seconds in a row. You could have the rockets on for 4.9s, off for 0.1s and on again for 4.9s.

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Re: Physics Question

Postby Anubis » Sun Nov 01, 2009 10:02 am UTC

stardek wrote:
{delta} wrote:In addition to that question, is the 5 second delay ONLY if they fire continuously for 5 seconds, or is it any time it accumulates 5 seconds of firing time there need to be a one second, no fire, delay?


The delay is only after firing for 5 seconds in a row. You could have the rockets on for 4.9s, off for 0.1s and on again for 4.9s.


Are you sure that's right? Because that makes the restriction pretty trivial. You can just periodically turn them off for an arbitrarily small amount of time and it will be functionally equivalent to a constant 19 second burn.

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Re: Physics Question

Postby edgey » Sun Nov 01, 2009 12:06 pm UTC

Yeah, in that situation you could just turn the rockets off for .0000000000000001 seconds and just treat it as if you never even stopped...

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Re: Physics Question

Postby skeptical scientist » Sun Nov 01, 2009 12:10 pm UTC

Well, without the max 5 second burn thing, here is what we get:

The total change in kinetic energy is equal to Mgh-W, where M is the mass of the rocket, and W is the work done by the rocket engines. The work W is equal to Mad, where a is the acceleration due to the rockets (20 m/s2), and d is the total distance over which they are firing. Here Mgh=M*10,000 m2/s2, and the kinetic energy at landing can be at most M*12.5 m2/s2 (the kinetic energy when speed is 5 m/s) so the work due to the rockets has to be at least M*9,987.5 m2/s2, so they must be exerted over a distance of 499.375 m. In order to minimize the time the rockets are firing (which serves to minimize both fuel use and total time, since the total time will be equal to the time the rockets are firing plus the time they are not, which must be within half of second of the time they are firing), we want to maximize the average speed over the time when the rockets are firing, so we want to fire the rockets at the last possible instant: when the ship is at height 499.375 m. So the optimal solution leaves the rockets off for 500.625 meters (10.006 seconds), and on for 499.375 meters (9.994 seconds), and uses slightly less than 10 units of fuel.

With the max 5 second burn thing, as you are interpreting it, we can get arbitrarily close to this by turning the rockets on for 4.997 seconds, then off for a nanosecond, and then on for another 4.997 seconds.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson

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stardek
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Re: Physics Question

Postby stardek » Sun Nov 01, 2009 5:19 pm UTC

Anubis wrote:
stardek wrote:
{delta} wrote:In addition to that question, is the 5 second delay ONLY if they fire continuously for 5 seconds, or is it any time it accumulates 5 seconds of firing time there need to be a one second, no fire, delay?


The delay is only after firing for 5 seconds in a row. You could have the rockets on for 4.9s, off for 0.1s and on again for 4.9s.


Are you sure that's right? Because that makes the restriction pretty trivial. You can just periodically turn them off for an arbitrarily small amount of time and it will be functionally equivalent to a constant 19 second burn.


My apologies, after rereading the question it seems there must be a 1s pause between any burst of the rockets, no matter the length.


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