The art of estimations is appealing abundant a physicist’s aliment and er. We all adulation a acceptable admiration problem. You ability additionally apprehend these alleged a “back of the envelope” calculation, or a adding on a napkin. The autograph average is meant to accent how little alertness goes into advancing the problem. The estimator can’t alike booty the time to acquisition a apple-pie breadth of paper.
But these calculations aren’t aloof for fun, they are additionally useful. One acceptable archetype is a catechism that I aloof afresh calculated: How big of a airship would you charge to lift a human?
Woah! Appropriate there bodies ability be saying: But wait! Is this a hot air balloon? Is this a hydrogen balloon? What actual is it fabricated of? The acknowledgment is “I don’t know, I don’t care.” The accurate ability of the admiration botheration is that you can appealing abundant avoid things that you don’t know. Yes, this takes some convenance to amount out what is important and what isn’t. With that said, let me alpha off with my assumptions and estimations.
Let’s begin. In adjustment for a animal to be aerial by a balloon, the airiness force charge be according to the weight of the human. The airiness force for any article is according to the gravitational force of the air (or water) displaced by that object. If I apperceive the body of air (I do), I can break for the aggregate of the balloon. Aloof to accomplish the calculations simpler, I am activity to use a cubical airship instead of a all-around balloon—they are aing abundant to the aforementioned admeasurement that it shouldn’t matter.
With an air body of 1 kg/m3 and a animal accumulation of about 90 kg, the aggregate of the airship would accept to be 90 cubic meters. Now I aloof charge to booty the cubed basis of 90—without a calculator because that would blemish the process. OK, cube roots are tough, so let’s aloof get a asperous value. I apperceive that 5 cubed is 125 and 4 cubed is 64. I’ll aloof breach the aberration and say that the cubed basis of 90 is 4.5. That agency the cube is 4.5 meters beyond or about 12 feet. The airship would accept a bore of about 12 feet.
OK, this acknowledgment is wrong. It’s amiss for lots of reasons. I didn’t use the actual appearance of the airship and I acclimated the amiss amount for the body of air (it’s absolutely afterpiece to 1.2 kg/m3). The bigger aberration is that I didn’t booty into annual the accumulation of the gas in the airship and I didn’t accommodate the accumulation of the airship material. But actuality absolutely appropriate isn’t the point of an estimation. With my estimation, I at atomic accept a asperous answer. I apperceive that this is activity to accept to be bigger than the bigger affair airship you can acquisition at a store. You aren’t activity to be able to alike grab a agglomeration of balloons from a assuming antic and use that to float away. It’s got to be a BIG balloon. Bigger than 12 meters across.
See? Wasn’t that admiration fun? I’ll do addition one, but let me aboriginal go over some of the key apparatus of a abundant estimation.
You accept to alpha with something. There has to be some appraisal or physics assumption that you alpha with. You can’t aloof appraisal the accumulation of the moon if you don’t apperceive annihilation about force or how continued it takes for the moon to apogee the Earth. In the archetype above, I aloof happened to apperceive the body of air because I’ve acclimated it before. I additionally knew the accord amid airiness and the admeasurement of an article (Archimedes’ Principle). After that, I would be appealing abundant stuck.
Approximate ethics are fine. Does it amount that I acclimated an air body of 1.0 instead of 1.2 kg/m3? Not really. It’s accurate that an air body of 1.2 kg/m3 would accord a abate admeasurement of the balloon—but not decidedly smaller. Remember that I’m not aggravating to account the exact admeasurement of the balloon. I aloof appetite to see if it’s a reasonable size. So, go ahead. Use a amount of 10 N/kg for the gravitational acreage (instead of 9.8 N/kg). Use a animal accumulation of 100 kg if that makes things easier. In the end, it won’t amount too much.
Estimation problems are array of like a asperous pencil sketch. It doesn’t accept all the exact capacity of a photograph, but it’s abundant to accord you an abstraction of what’s activity on.
Pick the simplest adaptation of the problem. If you appetite to appraisal the thermal radiation from a cow, you charge to apperceive the apparent breadth of said cow. That’s complicated, so let’s accomplish it simpler. What about a all-around cow? (classic physics joke) Alike admitting beasts aren’t absolutely spherical, they are aing abundant to actuality all-around for this problem. Also, the apparent breadth of a apple is appealing accessible to calculate.
If you don’t apperceive a value, at atomic get a reasonable range. But wait! What if I don’t apperceive the body of air? How could I do the airship problem? OK, let’s get started. What is some body amount that you DO know? What about water? That has a body of 1000 kg/m3. Should air accept a college or lower density? Yes, lower. But how abundant lower? 100 kg/m33 seems too aerial still. What if you acclimated 10 kg/m3? That’s appealing high—but it’s at atomic a start. Also, the body is acutely greater than 0 kg/m3. How about a ambit from 0.1-10 kg/m3. See, at atomic we accept article to assignment with now.
Check your answer. If you are ciphering the admeasurement of the airship and your acknowledgment is 4.5 kilograms, it’s apparently not correct. Units amount and you should accomplish abiding your acknowledgment has the actual units. There are added checks too. Did you get a abrogating amount for the airship diameter? You ability appetite to amend that estimation. But units are additionally your friend. If you don’t apperceive the accord amid baptize breeze amount and baptize speed, you can apparently amount one out aloof by authoritative abiding the units assignment out.
Here is addition fun admiration botheration (one that has absolutely appear up added than once). Why would you ask this problem? Suppose I appetite to analysis article falling at terminal velocity. The accessible affair is to bead an article such that it moves at terminal velocity. But how aerial would that be alone from? Let’s do it.
Here are my starting assumptions and estimates.
When an article avalanche from blow (in air), there are about two armament acting on it. There is the bottomward cull from the gravitational force, and again as the article increases in dispatch there is an advancement air attrition force. As continued as the gravitational force is greater than the air attrition force, the article will access in dispatch as it moves down. Terminal dispatch is that point in the motion area the two armament are counterbalanced to accomplish the article move at a connected velocity.
It’s absolutely appealing difficult to archetypal the motion of an article with air resistance. The botheration is that the force (and appropriately the acceleration) depends on the velocity—but the dispatch depends on the acceleration. So let’s accomplish it simpler. I will aloof avoid the air resistance.
If there is no air resistance, the animal will abatement with a connected dispatch of 10 m/s2. I can now account the time it takes to get to a dispatch of 50 m/s. Back the article increases in dispatch by 10 m/s every second, it would booty 5 abnormal to ability this dispatch (using the analogue of acceleration).
What about the ambit though? The article isn’t affective at 50 m/s the accomplished time—but who cares? Let’s aloof accept that anyway. If it moves at a dispatch of 50 m/s for 5 seconds, it would biking a ambit of 250 meters. So, this is a acceptable starting answer. You charge to bead it at LEAST from a acme of 490 meters—probably alike college back it won’t absolutely be activity that fast.
I rarely account being on an envelope or atom of cardboard (though I still do that sometimes). What I do instead, and it’s aloof as quick, is actualize a quick after adding with python. This is abnormally accurate now that I can run python being on my acute phone. I can crank out article cool quick—it’s still an estimation, but it’s a bigger estimation.
For the archetype of a falling human, I can now accommodate the air attrition force and accomplish a artifice of dispatch as a action of time. Here’s what I get (I will skip best of the capacity for now). Aloof columnist “play” to run the adding and you can adapt the cipher after breaking stuff.
Notice that by bottomward from a acme of 250 meters, the animal doesn’t absolutely ability terminal velocity—but it’s a appealing nice estimate.
I can’t leave you after some added admiration questions. Here are some questions for you to comedy with. Accept fun.
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