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What extra data do you have: there are too many variables to be accurate now.
Well, I need an approximate answer, variables are what you can think of. I am first considering that the heat is equal to the work that is done by the person, meaning first we have to take into account the force needed for acceleration to a certain speed, combined with the force needed for overcoming the air resistance and then we add to those two just the force needed to keep going (overbearing the air resistance). This multiplied by the corresponding distance travelled and then we add the constant body heat that is emitted over the time of running. Is my logic right? If so, my problem is that i can't express it algebraically.
You want to look the energy burned by the muscles and compare that to the energy dissipated as drag?
More like the combination of the two, the total energy.
I would estimate it theoretically. http://www.grc.nasa.gov/WWW/k-12/airplane/drageq.html I don't think you're going to get anywhere looking at energy expenditure in the body. Drag contributes very little to the energy required to run. Running is essentially an extended series of shallow jumps, and almost all of the energy expended goes into overcoming gravity, and accelerating and stopping the motion of your legs, with a bit more going to the bellows actions of your lungs, pumping air in and out, and the pumping action of your heart.
Blah, you're right. Seems like a lost cause? Because this was one of the questions in a Cambridge interview and I wanted to know how the problem is solved.
Well, having done lots of interviews in my day, I would suggest they were less interested in whether you knew the right answer than in whether you could (1) reason your way towards how you would find the right answer, and (2) be able to make very crude estimates of the rigth answer. For example, if I were the interviewer, I would have looked for you to say something like "It can't be very much, because human beings are not streamlined, the way fish or birds are, so there can't be much benefit to be gained (in extra running speed for the same energy expenditure) by reducing air resistance. Additionally, I've never seen anyone in the Olympics try to minimize air resistance by (for example) running head down to minimize the area he exposes to the oncoming wind, or wearing special clothes to reduce drag." You could then go on to say that you know the field that contains the science you need is aerodynamics, which is closely related to hydrodynamics, and that you know that generally speaking the drag exerted by a fluid (air or water) on a moving object is directly proportional to the frontal area exposed to the fluid flow (so blunt objects experience more drag than sharp or streamlined objects), as well as to some power of the forward velocity (which is why air resistance grows sharply with velocity, and while runners don't bother with streamlining themselves, bicyclists do). You can also observe that it must be a strong function of the density of the fluid, because fish and human swimmers make an effort to be streamlined even though they are moving slower than runners in air -- so the drag of the water must be much higher at a lower velocity. Finally, you could observe that the nature of the drag probably changes with some combination of velocity, density and viscosity of the fluid, because birds flying don't do the same thing to minimize their drag as do fishes. For example, geese fly fast, and are streamlined, but they also flap their wings vigorously and rapidly. On the other hand, dolphins swim fast, and they are also streamlined, but they do NOT flap things -- instead they wiggle their tails in some smooth way. You would surmise that when you move through air fast you can afford to do rapid back-and-fortth movements, like flapping, but when moving through water you need to be more smooth. What you would have put your finger on here is the Reynolds number, which indeed broadly characterizes the nature of motion through a fluid. In the end, it sounds like they were probing how much hydrodynamics you knew, or could reason out on the fly. Tough question, since it's a tough field, and few people study it at a young age.
Thank you so much. You sure know how much you're helping me, so again, thanks. :)
My pleasure. Keep in mind that your daily experience already tells you quite a lot about underlying physics principles, so it's often fruitful to consider it with a "physicist's eye" and see what you can conclude. Also: generally when people are interviewing you for admission to a program, or a job offer, they are NOT looking to see if you already know lots of stuff you'll need or use with them. They're going to teach you that anyway. What they are often looking for is (1) how observant you are, and how effectively you use your observation (i.e. how good a reasoner you are), and (2) how flexible you are -- how readily you can derive useful results from unexpected or nonideal sources of information, and (3) how determined and "plucky" you are, how resilient in the face of defeat or challenge, how resolutely you look for secondary paths to success. The reason is that all three of those things are much better predictors of how well you do at a complex challenging task (like a college or graduate degree, or challenging job) than just the amount of knowledge you have stashed up in your head already. I realize this is not necessarily the way you're taught at school, where (unfortunately) you're often encouraged to just keep your head down, do as your told, memorize accepted wisdom, and not do too much thinking for yourself. That's an unfortunate aspect of modern mass-production education.
Here's a classic anecdote on the issue: http://naturelovesmath-en.blogspot.com/2011/06/niels-bohr-barometer-question-myth.html
I know, a lot of the school hours are spent in useless discussions about Big Brother, just imagine... And the sad thing is that the teacher is a very intelligent man, yet he does not realise that by criticising vulgarity he actually makes it popular. Nevermind, I have for a long time now been led by "Never let schooling interfere with your education".
Excellent answer Carl, sorry mandja for not getting what sort of answer you wanted.
Thanks to all of you :)