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We have the following objects:

  • a 80 kg person
  • a rope of negligible weight
  • a balloon filled with helium, which can lift for around the same weight, 80kg.

My question is, which of the following outcomes will occur (without considering weather conditions)?

  1. The balloon can actually lift a bit more than 80 kg.
  2. The balloon cannot lift more than 79.9 kg

Most importantly, suppose this person manages to do a 10 meters jump, on the way down, would he accelerate too fast for a "safe" landing?


enter image description here

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Could you provide the details of this "jump"? How does he jump? Does he step off, or jump upwards and fall? Or jump to the side? –  Manishearth Dec 6 '12 at 10:38
    
a simple jump upwards –  john smith Dec 6 '12 at 10:40
    
Not that simple. Depends on how hard he jumps. Though, since you want a ballpark answer, I guess it's fine... –  Manishearth Dec 6 '12 at 10:51
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The confusion is directly and artificially inserted to the question. In the third bullet, we're told that the balloon can lift about 80 kg. It deliberately says nothing whether it can lift more than 80 kg or less than 80 kg. Then we're asked whether the actual number is less than or more than 80 kg or 79.9 kg. What is it supposed to mean? You're asking about exactly the same quantity that you deliberately described inaccurately, so how can we tell you the accurate value? –  Luboš Motl Dec 6 '12 at 11:15
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I think this question is potentially interesting, but in its current form it is completely unclear what you want to know. Can you clarify the thing mention by Lubos and maybe add a sketch? –  Bernhard Dec 6 '12 at 11:23
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closed as not a real question by Manishearth, David Z Dec 6 '12 at 18:31

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2 Answers

Both object are having same capacity. So earth gravity effects on person acceleration 1.both are stay at constant position.

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Suppose the the gravitational force of the person+ballon $F_g=-mg$, just exceeds the lift force $F_{lift}$, such that $$F_{res}=F_g - F_{lift} < 0 $$

Then in the case of a solid rope (=rod), this will be just the same as a reduced gravitational force

$$g'=g - \frac{F_{lift}}{m}$$

And the math stays the same, so the velocity during landing will be the same as during take off.

In this case, the flexible rope does not really changes things in my opinion. As soon as the guy jumps, the downward force on the balloon is gone, such that it rises quickly to pull the rope straight again, reinstating the small downward force on the guy. However, also the upward force on the guy is gone temporarily, so I don't think he will be able to jump 10 meters. The landing velocity will only be smaller than the take of velocity, as the downward acceleration is becoming less on descent.

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Correct neglecting air resistance. Perhaps a little less if you allow for it. Or notably less as the ballon is going to be pretty big. –  dmckee Dec 6 '12 at 16:03
    
Haven't you forgot added mass? en.wikipedia.org/wiki/Added_mass Though I'm too lazy even to estimate how important it is for the case. –  Yrogirg Dec 28 '12 at 8:45
    
@Yrogirg I neglected air resistance already, which is generally more important than added mass. –  Bernhard Dec 29 '12 at 8:29
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