These puzzlers are to test your resolve. See if you can solve them before reading the solutions.
These Puzzlers are in the form of an experimental project, that is, you will need to design your own experiment in order to solve these puzzlers.
SpaceNews: Mad Scientist Solves Rocket Problem for NASA Using, of All Things, a Water Rocket!
(Because of budgetary limitations, NASA must solve this problem with a minimum of expense. Hence they have only enough money for some simple tools, nothing fancy like a billion dollar, triple-frequency, laser ranging instrument.)Can you devise a strategy for measuring the maximum altitude attained by your water rocket using only the tools provided: a protractor, some tooth floss, a cardboard core from a spent toilet paper roll, a pencil, some graph paper, and, just maybe, a slide rule.
The second part of the problem is to maximize the altitude achieved by your water rocket. I suggest making a table of various water quantities and how high each one went. In order to compare only the amounts of water, always pump the rocket the same number of times, say 20 times, before launching it. For instance, fire one launch with no water in the rocket, just pumping it 20 times, and record how high it went. Fire another one with the rocket filled completely to the top, record how high it went. Then try 1/4, 1/2, 3/4 full rockets and record all of your results. Compare your data and choose the best amount of rocket fuel (water), that is, the amount that yields the highest altitude.
Did you notice that neither the zero rocket fuel (water) test nor the completely full rocket test achieved the highest altitude? One might expect that if some water is good, then wouldn't more water be even better! So, why doesn't the completely full rocket fly the highest? Can you come up with a reasonable explanation for this apparent incongruity?
Here is the design and experimental for the solution:
And, for those who wish to review some of their differential equations and fluid mechanics, an extension to the water rocket altitude puzzler is to explain why the water rocket does not work as well when the fuel is just plain old air...especially since according to fluid dynamics the thrust of a rocket does not depend upon the density of the effluent, so both air and water should provide the same thrust! (And yes, this is a shocking development.) So, if water and air both provide the same thrust, why does the rocket not work very well with only air?
Note: If you would like to understand more about the physics behind the flight of the water rocket, the "Water Rocket Thrust" Example of the Fluid Mechanics Chapter derives the equations for the thrust of the water rocket, the speed at which the water is expelled from the nozzle, and what is known as the Ideal Rocket Equation that explains the excess velocity attained by a rocket that continuously looses mass to the efflux. In this Example we also explain why the water rocket works so well with water as the 'fuel' but not very well with air alone. Here is the PDF file for this Example:
Attach two cups to the end of a wooden plank. One cup holds a baseball, the other is empty. Leaning the plank diagonally into the corner of a wall and the floor, drop it. The baseball magically transfers from its initial cup to the second cup. Wow! Why?! How?!
This project exits on several levels... First of all, there is the design and building of the rotating plank itself, secondly there is the explanation of why the ball separates and lands in the cup in the first place (in the simplest explanation the ball should just fall right along with the plank and thus should not separate from the cup), and lastly there is the full theoretical analysis of this problem and explanation for why you should not attempt to have the plank drop farther than about 24 inches...
When you have solved this problem, you will then also know why chimneys and tall radio towers break in half when they fall over...
Hint: The moment of inertia of a plank is:
With the moment of inertia you can calculate the angular acceleration of the plank from which you may then obtain its linear acceleration... This will allow you to calculate the maximum angle above which the experiment will not work (roughly 36 degrees).
Here is the Rotating Plank Experiment and Analysis:See the interesting photos: http://gizmodo.com/5304233/entire-new-13-story-building-tips-over-in-shanghai/
Using whatever common tools and items you have around the house, design an experiment to measure the local acceleration of gravity.
There are a number of ways of determining this ... for instance, you might think about using a pendulum, or how about an inclined plane?
For example, since we had three individuals which allowed us to make 6 measurements for each experiment, we decided to measure the location at six different positions as a roller rolls down an inclined plane. We used a plastic folding table as an inclined plane, two CDs and a brass tube for the roller, a piano metronome to measure time, Post-it notes to mark the position of the roller for each tick of the metronome, and a yard stick to measure the distances.
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We measured 5 distances each experiment and repeated the experiment 10 times. We then statistically analyzed the data to calculate the best fit curve in order to derive the local acceleration of gravity with error bars.
Because our model and system of measurements did not fit neatly into a scientific calculator's ready-made statistics program, we wrote our own simple computer program to perform the chi^2 fitting of our data to our model.
Here is the design of our experiment and the statistical analysis of the data:
A fellow rancher wanted to determine the area of one of his oddly shaped pastures...
A Calculus-based Introduction: [You may skip to Heron's Formula if you wish.]
Before we get started with this Surveying Thinking Problem, allow me to provide a short alternative introduction for those of you who have already studied Calculus. [You may skip to Heron's Formula if you wish.] This alternative describes what I think is one of the more remarkable relationships of Calculus, known as the First Fundamental Theorem of Calculus, that makes a connection between the derivative and the integral. In a fashion, it says that integration "undoes", or is the inverse of, differentiation, and vice versa. In one dimension, the First Fundamental Theorem of Calculus equates an integration of the derivative of a function over an interval to the function value at a single point (an infinite summation in 1-D equated to a single value in 0-D). When extended into two dimensions, the analogous theorem is known as Green's Theorem. It equates a double integration of the partial derivatives of a function over a region to a line integral of the function along the boundary line of that region (summation in 2-D equated to a sum in 1-D). And in 3-space this concept generalizes to the so-called Divergence Theorem that equates a triple integral of the divergence (derivative) of a function over a volume to a double integral of the function over the boundary surface of that volume (sum in 3-D equated to a sum in 2-D). And the Divergence Theorem is a special case of the more general n-dimensional Stokes's Theorem over differential forms.
From an alternative point of view, the Divergence Theorem in 2-dimensions is equivalent to Green's Theorem, and Green's Theorem in 1-dimension is equivalent to the First Fundamental Theorem of Calculus.
The recurring theme in the above dimensional generalizations is that a summation (integration) of the appropriate derivative of a function in N dimensions is equated to a summation (integration) of the function in N-1 dimensions. This shocking decrease in dimension leads to some remarkable results, as we shall see in a moment. But for now, just remember the recurring theme: a sum (integral) in N dimensions is equal to a sum (integral) in N-1 dimensions. (It is this theme that produces the measurement without "touching" idea.)
Now on to describe the Land Surveying Problem...
Heron's Formula:
Say we wish to determine the area of a triangle. The standard high school geometry formula is Area=(1/2)bh where b is the length of the base and h is the length of the altitude to that base. The altitude is an "internal" measurement for the triangle, it requires "going inside" the triangle in order to find its value. [Or some logically equivalent measurement to "going inside", such as the measurement of the exterior angle followed by subtraction from 2pi -- this is an indirect equivalent to "going inside".] Other formulae exist that utilize trigonometric functions of the angles of the triangle, for instance, Area = (1/2)(ac(sin B)+bc(sin A)). Once again, an angle is an "internal" measurement, in that one must sweep across the interior of the triangle in order to measure the angle. If we were to employ Calculus to measure the area of a triangle, once again we would divide up the interior into rectangles and sum the areas of those rectangles. In the limit of an infinite number of infinitesimally narrow rectangles, this sum becomes the triangle's area. So, even the standard area measurements of Calculus require "internal" measurements.
In fact, all triangle area formulae require some form of "internal" measurements, with the single exception of what is known as Heron's Formula. Heron's Formula gives the area of a triangle only requiring the lengths of the triangle's sides, no angles nor other types of internal measurements are needed. Now this should be surprising to you on two fronts. First of all, we are measuring the interior of a triangle but without actually going "inside" the triangle. It is a form of measurement of the interior without actually having to make a measurement over said interior, that is, it is measurement without "touching". And secondly, we can draw two triangles having entirely different areas but the same perimeters. Since Heron's Formula relies on the perimeter length, how can it possibly determine the area of a triangle if two triangle's with the same perimeter lengths have entirely different areas?
Nevertheless, it's true, we can measure the area of a triangle without any "interior" measurements. Say we wish to survey a pasture, but we don't have a digital laser theodolite at our disposal. The theodolite would measure the angles of a triangle with great accuracy, but without one we would be reduced to using a dime store protractor. Our angle measurements would therefore be prone to fairly large errors. And when one makes a measurement on an angle that is in error by say 1 degree, the pie-shaped area corresponding to this angle error when we get out to a distance of 600 yards is rather significant. Instead of the theodolite, we rather have a perambulator or perhaps even a laser rangefinder used in archery target shooting. Can we just measure the sides of the triangle using the perambulator or laser rangefinder and still be able to calculate the area?
Sandy holding a target for the laser rangefinder. And below are the measurements made... (notice that many of the angles are not 90 degrees)Here are the distance measurements...Puzzler: From the above measurements, compute the area of the pasture.
The PDF file, , for this problem includes the Heron's Formula approach to surveying a pasture. The original proof of Heron's Formula was by means of Euclidean geometry is also given. It is a truly Sherlockian proof, apparently roaming around getting nowhere fast until the very last step when everything mysteriously falls into place. We include this 24 page romp through much of Euclidean geometry. But we also derive Heron's Formula using Vector Algebra, a strategy that is both concise (2 pages) and provides intuition into why Heron's Formula is true in the first place, something that is not forthcoming from the Euclidean proof.
And now for the real shocker --> Okay, so we now understand that the interior area of a triangle can be determined without having to make any measurements on the interior of said triangle, but would you be shocked to learn that this statement is true not only for triangles with straight sides but also for most any general closed curve? Yup, consider a small, irregularly shaped, pond. You can determine its area simply by making a 1-dimensional measurement along the shore of the pond! You don't have to make any "interior" measurements to find its area. This is where the Calculus-based Introduction comes into play: this stunning result is a consequence of Green's Theorem, the theorem that equates a 2-D integration (area computation) to a 1-D integration (length computation). And the following mechanical device, known as a polar planimeter, implements Green's Theorem in gears and measures areas by following the 1-dimensional border of those 2-dimensional areas. (In other words, Green's Theorem allows an inherently 1-dimensional measurement to calculate an inherently 2-dimensional quantity, and this theorem is implemented mechanically using just a few gears.)
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At the request of Napoleon III, Leon Foucault hung a pendulum from the top of the dome of the Pantheon in Paris in 1851 in order to demonstrate to the world that the Earth turns. In the 1995 Foucault's Pendulum was re-hung at the same location. It certainly looks tall, so can you measure the actual length of Foucault's original pendulum demonstration using just what you have with you as a tourist visiting Paris?
This photograph is a floor level view of Foucault's Pendulum swinging back and forth at the Pantheon in Paris. By averaging over a number of swings, we determined that it takes over 16.46 seconds for the pendulum to make a single swing.How long is Foucault's Pantheon Pendulum?
Hint: Remember your watch...
Here are the measurements and the calculations, including the derivation of the equations, to determine the length of Foucault's Pendulum:
[This is an Advanced Puzzler. It requires an understanding of the properties of Continued Fractions as well as knowledge of Bernoulli Trials and the moments of the Binomial Distribution, some Probability Theory, in other words. It is a tough go, but the rewards are correspondingly great -- an understanding of how one can prove that an experiment has been faked.]
in 1901 Lazzarini reported the following results ...
(Work still in Progress...more to come...)
Say you don't have an expensive chronograph, then how can you determine the speed of your arrows?
(Work still in Progress...more to come...)
I know, it is hard enough for us to learn something and we have brains, but have you ever thought about how you might design a machine so that it too learns to solve a problem all by itself? Here is just such a machine...
(Work still in Progress...more to come...)
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