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Physics Q&A Game

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  1. Apr 13, 2005 #1

    Gokul43201

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    A Q&A game is simple: One person asks a relevant question (it can be research, calculation, off-the-top-of-the-head, anything as long as it is a physics question) and other people try to answer. The person who posts the first correct answer (as recognized by s/he who asked the question) gets to ask the next question, and so on.

    I'll get this rolling with a simple back-of-the-envelope calculation question.

    It is proposed to make a man powered helicopter, with a rotor 10 m in diameter. Assuming that the rotor blows a cylindrical column of air uniformly downwards, the cylinder diameter being the same as the rotor diameter, and the weight of the man plus machine is 200 kg, calculate the minimum mechanical power (in watts) that the man must generate, if he is to remain airborne. (take density of air ~ 1.2 kg/m3) Is the system feasible ?

    (show, in a couple of lines, the essential steps in the calculation)

    If a sufficient time passes with no correct answer, then the best attempt determines who goes next.
     
    Last edited: Apr 13, 2005
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  3. Apr 13, 2005 #2

    Galileo

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    Great initiative Gokul!

    Here's my attempt. Call the radius of the rotor r.
    The power generated is the kinetic energy transferred to the air per second.
    The mass of the air column is the density of the air times it's volume, so

    [tex]K=\frac{1}{2}Mv^2=\frac{1}{2}\rho \pi r^2 vt v^2[/tex]
    [tex]P=\frac{dK}{dt}=\frac{r^2\pi}{2}\rho v^3[/tex]

    The momentum imparted to the air is:

    [tex]P_m=\rho \pi r^2 v^2t[/tex]
    so the force that keeps the copter floating and should equal mg is:
    [tex]\rho \pi r^2 v^2=mg[/tex]
    which we can solve for the speed of the air particles.
    We find for the power:

    [tex]P=\frac{r^2\pi}{2}\rho \left(\frac{mg}{r^2\rho \pi}\right)^{3/2}[/tex]
    Plugging in the numbers, we find about 4,5 kW.
    I had to look up some power consumption data. If a human runs at 24 km/h, the power consumption is about 1.7 kW. So it's not a feasible construction (not for me at least).
     
    Last edited: Apr 14, 2005
  4. Apr 13, 2005 #3

    Gokul43201

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    Nice one Galileo !!

    World class sprinters and trained athletes can deliver close to 5 kW for no more than a minute. I certainly can not sustain more than about a kW for any reasonable length of time.

    Your turn...
     
  5. Apr 13, 2005 #4
    Gokul and Galileo,

    Notice though that the required power goes as 1/r. This makes sense because you're getting your momentum with more mass and less speed, and that reduces the energy which goes as speed squared.

    So maybe with 50m rotors (made out something with a very high strength/weight ratio) you could do it. Of course your ideal model of the air column moving down monolithically will probably fail at low speed. Oh, well!

    PS Gokul, be aware that your Q&A game could be abused by students wanting a quick solution to a homework without doing any work themselves! Come to think of it, maybe that's what you were doing. :wink:
     
  6. Apr 13, 2005 #5

    Gokul43201

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    Point taken, on the abuse possibility (I'll not comment on the snide insinuation just yet). I'll leave it to the mentors to make that judgement.

    As for the 50 m blades : you find me the material, and we're in business ! :biggrin:
     
  7. Apr 13, 2005 #6

    Galileo

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    I think homework questions posted here are easily detected. Such are usually not the kind of questions you want to post here.

    As a result, it took me a little time to think up a nice question (that I have to solve myself), so I actually should have taken that into consideration before posting a solution :redface: , but anyhoo:

    There's an astronaut going for a space walk and enjoying the awesome sight (who wouldn't).
    Just when he's ready to return to his spaceship he notices the line that connected him to the spaceship has broken!! This line was supposed to bring him back aboard the ship. The astronaut has only 5 minutes of oxygen left in his tank and he fears for his life. If something can be done, it has to be done quickly.
    The astronaut realizes he has a small fire extinguisher on his spacesuit which he can use as a thrustrocket. He immediately grabs the extinguisher and wants to start spraying.

    The astronaut is now 120 m away from his spaceship and has a neglegible velocity with respect to the ship. His mass (with suit, but without extinguisher) is 94.5 kg. De mass of the contents of the extinguisher is 5.0 kg and the mass of the empty container is 5.5 kg. The contents are sprayed outwards with a speed of 10 m/s, regardless of the remaining content. After 2 minutes of spraying, the container will be empty.

    Will the astronaut arrive at the spaceship before his oxygen supply runs out?
    Can he save his hide from this perilous predicament? Stay tuned!
     
  8. Apr 13, 2005 #7

    Gokul43201

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    I think it may be important to specify that the extinguisher is attached to the spacesuit, and can not be removed. Else, there's an easy cheat.
     
  9. Apr 13, 2005 #8
    [tex]v_f = v_i + v_{ex}*ln(m_i/m_f)[/tex]

    [tex]v_f = 10ln(\frac{105}{100}) = 0.487m/s[/tex]

    At t = 120s he will have travelled 120x0.487= 58.55m

    At t = 600s when his gas is empty he will have travelled 600x0.487 = 292m.

    He makes it with time to stop for donuts.
     
  10. Apr 13, 2005 #9

    Galileo

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    I`m sorry, your [itex]v_f[/itex] is roughly equal to the final velocity of the astronaut, so you cannot use the distance formula for constant velocity.
    Also, there is 5 minutes of oxygen left, not 10.
     
  11. Apr 13, 2005 #10
    I have an idea for the solution:

    From the law of conservation of momentum

    [tex]m_{e}v_e=(m_a+m_c)v [/tex]

    Deriving with respect to t we find:

    [tex] \frac{d(m_e v_e)}{d t}=m_aa+\frac{dm_c}{d t}v+m_ca [/tex]

    We get an ODE which is easy to solve, since we know that

    [tex]\frac{dm_e}{d t}=-\frac{dm_c}{d t}=5 kg/120s[/tex]
    [tex] v_e=10 m/s[/tex]

    The total time required to transverse distance s is

    [tex]t=\frac{s-\int_{0}^{2min} v(t)dt}{v(120s)}+120s [/tex]

    P.S.
    Sorry I didn't do any numerical calculations. My MS Calculator crashes everytime I try to start it, and I don't know where my pocket calculator is. :frown:

    P.P.S.
    I read your question again and made the necessary correction.
     
    Last edited: Apr 13, 2005
  12. Apr 14, 2005 #11

    Galileo

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    Good try Berislav.

    I can make the calculations for you if you tell me what numbers to plug in. Since you haven't got a formula for v(t) I can't solve your equation for t yet.
     
  13. Apr 14, 2005 #12
    For [itex] v(t) [/itex] I get:

    [tex]v(t)=\frac{5/120v_et}{-5/120t+10.5+m_a}[/tex],

    since [tex]m_c(t)=10.5 kg-5/120t [/tex]
     
    Last edited: Apr 14, 2005
  14. Apr 14, 2005 #13

    Galileo

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    Ok, well you'll have to integrate it yourself. There are some internet tools to help you with it if your require it: http://integrals.wolfram.com/
    Online calculators also exist, just goooogle it up.
     
  15. Apr 15, 2005 #14

    Galileo

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    Anyone?

    Too hard? Too easy? I`ll start posting mild hints tonight (on my clock) if no one answers.
     
  16. Apr 15, 2005 #15
    Wouldn't average velocity for the first two minutes work?

    [tex] v_avg = 0+0.487 / 2 = 0.2435/s [/tex]

    [tex]0.2435m/s * 120s = 29.22m plus[/tex]

    [tex]180 * 0.487 = 87.66m[/tex]

    = 116.88m. I'm not sure though, my algebra based physics skills are no match for you guys.
     
    Last edited: Apr 15, 2005
  17. Apr 15, 2005 #16

    Gokul43201

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    Can I suggest that you give it about 24 hrs before releasing hints ?
     
  18. Apr 15, 2005 #17

    Chronos

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    Try using seconds as the time unit and see what you get. All the relevant constants defining work are so based.
     
  19. Apr 15, 2005 #18
    Integrator gives me:

    [tex]s(t)=\frac{5}{120}v_et \log(10.5+m_a-0,0416667t)[/tex]

    EDIT:
    No, scratch that! I meant at t=120s this is 100m. So,

    [tex]t=\frac{s-\int_{0}^{2min} v(t)dt}{v(120s)}+120s [/tex]

    is 160 s. So, he makes it. It seems that mulitplication and division elude me. :cry:

    P.S.

    Sorry I posted this so late. :frown:
     
    Last edited: Apr 15, 2005
  20. Apr 15, 2005 #19

    Galileo

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    Whozum. Taking the average velocity gives a good approximation, but from your anwer you cannot answer with certainty whether the astronaut makes it or not.

    Berislav. You are close (i.e. on the right track), but there are some errors in your calculations.
    (160s is off, it would mean he'd make it easily and I can guarantee the outcome will be a close call)

    I wanted to post some hints at this time, since it's about two days after posting the problem, but at Gokul's request I'll wait a bit and post them in the morning (about 8:00 GMT) if no correct answer is in yet.
     
  21. Apr 15, 2005 #20
    Can I waive my ability to solve this problem and request you message me the math needed to solve it?
     
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