Real World Physics Problems

Inspired by Rightwingprof's post on real world business problems that aspiring business majors need to know how to do, I'm going to give you a simple example of a real world phsyics problem that an aspiring science/engineering major will be expected to do. Actually, it's not real world, it's a greatly simplified real world problem -- real world problems are much more difficult:

The positions of a particle and a thin (treat it as being as thin as a line) rocket of length 0.280 m are specified by means of Cartesian coordinates. At time 0 the particle is at the origin and is moving on a horizontal surface at 23.0 m/s at 51.0°. It has a constant acceleration of 2.43 m/s2 in the +y direction. At time 0 the rocket is at rest and it extends from (−.280 m, 50.0 m) to (0, 50.0 m), but, it has a constant acceleration in the +x direction. What must the acceleration of the rocket be in order! for the particle to hit the rocket?

You only need one physics equation to solve this problem: The distance formula.

The formula to determine the distance traveled by a moving object during a given time is:

distance = initial distance + initial velocity x time + 0.5 x acceleration x time²

Can you solve the problem?

All you need to solve the problem is the above formula and the math you should have learned in high school.

Let me give you a hint: The particle will collide with the nose of the rocket when it reaches the height (50.0 m) of the rocket since the rocket is flying level. How far along the x axis is the rocket at that point and how long did it take to get there? What is the acceleration at this time. What if the rocket hit the tail of teh rocket instead. What is the rocket's acceleration now.

No! w you know the steps to solve the problem. Now get to it.

Can't do it? Let me make it easier for you by telling you every step of the solution.

1. Decompose the velocity of the particle into its x and y components. You need to know simple trigonometry and algebra to do this. Here's another hint: The sine of an angle equals the opposite side divided by the hypoteneuse. The cosine of an angle equals the adjacent side divided by the hypoteneuse.

2. Now use the distance formula and determine how long it will take for the particle to reach the height of the rocket. You'll need to know how to solve a quadratic equation to do this.

3. Now determine how far along the ground the particle will be when it reaches the height of the rocket.

4. Using the distance formula again determine the acceleration of the rocket if the particle were to strike the nose. This step requires simple algebra to solve.

5. Using the distance formula again determine the acceleration of the rocket if t! he particle were to strike the tail. This step requires simple algebra to solve.

6. The answer is that the acceleration of the rocket must be between the aceleration of the nose and the acceleration of the tail when the particle strikes the rocket or else it misses.

Ok, I've set-up the problem for you. I've conceptualized the problem for you. I've given you the physics. All you need to do is to use your high school math to solve.

Go ahead give it a whirl. It's still not an easy problem to solve is it? Especially if your algebra and/or trig skills are rusty. Try guessing and checking your way to the solution. I dare you to try.

If you're uncertain how to calculate an arcsine, how distracted do you think you'll be having to shift gears and ponder that while the physics problem is waiting to be solved. Or, maybe you are a real smartie and know how to derive an arcsine from first principles, go ahead work it out. The physics! problem will still be waiting when you're done. Gee, I bet t! hat was distracting.

Now we get to the quadratic equation. Remember how to solve one of those off the top of your head? That means you have to derive it yourself or go look it up. The physics problem is getting impatient now. Do you even remember where you were in the solution. What's the next step? What are you trying to solve anyway using the quadratic formula? What do you do with the extra solution you get when you finally solve the quadratic equation? Now all you have to do is use the distance formula to solve for three different variables. To do that you'll need to know how to manipulate algebraic expressions with some skill. Are you confused. Does your brain hurt yet?

This is a simple college level physics problem the likes of which is going to confront every aspiring science and engineering student in their first year of undergrad. It gets a lot tougher than this real quick.

This is one of the first gatekeepers students will meet. ! Most will not get through the gate. Most will not get through the gate because they did not have the domain knowledge in math to enable them to quickly solve the hundreds of physics problems they'll need to solve in order to acquire the domain knowledge in phsyics in order to pass their Physics course.

Once students have the physics domain knowledge, solving problems like this is simple and requires little cognitive ability. What seems like an insurmountable problem to the novice, requiring enormous cognitive effort, turns out to be a simple problem for the expert. One that can be easily broken down into simple steps that can be solved using one formula and some high school math.

Here's another real world question. Can you determine whether a student scoring at the advanced level of the 11th grade NAEP or any NCLB state exam knows the math necessary to do my real-world physics problem.

Answer: No. There is nothing on either the NAEP or! any state exam even remotely approaching the difficulty level! of math needed to do the problem. Less than 10% of students perform at the advanced NAEP level.

Solve using Distance Formula

#13 Ratio and Proportion

Ratio and Proportion

The probability that 901 coins have total value $100

Here's a cute little problem from Reddit: Tough question for you guys. Let's say you have 901 coins that come out to exactly $100. What are the odds? (Also here.)

Everyone there who gets a solution is assuming that all the possible coins are equally likely, which isn't a reasonable assumption. Years ago I looked at the density of money, where I used a model in which I get back from each transaction n cents with probability 0.01, for n = 0, 1, ... 99; furthermore I always get back the smallest possible number of coins. The only coins allowed are pennies, nickels, dimes, and quarters (worth 1, 5, 10, and 25 cents respectively).

As I calculated before, if I! make 100 transactions, and I get each number of cents back exactly once, I'll get 200 pennies, 40 nickels, 80 dimes, and 150 quarters. This is a total of 470 coins, and worth $49.50. Thus the "average coin" is worth 495/47 = 10.53 cents; 901 coins are "on average" worth $94.89. The value $100 isn't that unreasonable.

So consider a jar with 901 coins, which are independent; they each have probability 20/47 of being a penny, 4/47 of being a nickel, 8/47 of being a dime, and 15/47 of being a quarter. The mean value of a coin is 495/47 = 10.53 cents; the variance is 238840/2309 = 108.12 "square cents".

The mean value of 901 coins, then, is 9489 cents; the variance is 93198 "square cents", so the standard deviation is 305 cents. (Everything here is rounded to the nearest integer.)

Invoking the central limit theorem, then, we say that the value of 901 randomly chosen coins is normally distributed with this mean and standard deviation. Th! e probability of having value exactly 10,000 cents is approxim! ated by the probability density function of this variable at 10,000; that's 0.000322, or 1 in 3101.

An exact answer is feasible -- but not worth computing, I'd say, because the error in the central limit theorem is surely much smaller than the error from the fact that this isn't a realistic model of what actually ends up in your change jar.

probability coins

Measurement Conversions , Significant Figures, and Scientific Notation

Monday: Quiz Tuesday over last two columns: Halogens and Noble Gases.
SWBAT: determine the number of significant figures in a measurement.
1.) CALM work for 25 minutes.-Homework if not done.
2.) Test: Classifying Matter, First 2 columns of periodic table, extensive vs. intensive properties, physical vs. chemical changes.
3.) Significant figures intro discussion: Different measurement tools give you different numbers of significant figures. Example: Your hand, a triple beam balance an analytical balance.
3.) Significant Figures lab stations, using image on pg. 67 as an example. 16 stations different measurements followed by worksheet. Go through an example before releasing to stations.
4.) Reading pages 66-67 in the textbook, using

Wednesday: Quiz Friday over Oxygen and Nitrogen Groups.
SWBAT complete: Scientific Notation, Conversion and significant figures calculatio! n problems.
Review: Number of significant figures in numbers practice on the overhead.
1.) Review scientific notation using a table of yes, no, and correct the nonscientific notation problems.
2.) Pgs. 70-71, Read through example problem then students work 5-6, and 7-8.
3.) Go over answers in class from 70-71.
4.) Go over table 3.1, 3.2, 3.3, 3.4, 3.5, and equations on pg. 77. Emphasize the need to understand and memorize table 3.2.
5.) Conversion problem stations. Known unknown and plan.
16 stations where students make conversions at the stations-not made yet.

Friday:
1.) Start with conversions go over a warmup in pairs.
2.) Density Demo with different liquids, water, detergent, oil, alcohol, shampoo,
3.) Density and Temperature Discussion: when to fill up on gasoline, basketballs, tires,
4.) Review scientific notation through density calculation gizmo online.
5.) The n CALM problems.

Tuesday: Goal: Make conversions between scientific units.
1.) Unit prefixes table of notes in notebook.
2.) Go over two examples using dimensional analysis. Dimensional Analysis around the room.
3.) CALM conversions go over volume calculations for CALM.

Thursday
Describe the development of the atomic theory.
1.) Boxes, Measuring Mass, Volume, Density and determine mystery shapes inside of the boxes (lead into atomic theory)
2.) ppt. for Atomic Theory: Greeks, Dalton, Rutherford, Thomson, Bohr. (Person, time, experiment, and conclusion, and picture of atom, graphic organizer)

Tuesday:
Reading Atomic Symbols Practice
Understand differences between atoms, explorelearning.com Element Builder Gizmo

Thursday:
Atomic Symbols Warm-Up
Relative Abundance Examples
CALM problems

How to do scientific notation problems

Multiplying and Dividing Fractions

Last week we talked about adding and subtracting fractions, which most of us know is a very difficult subject for students. This week I want to add to this discussion and give some helpful websites to help students multiply and divide fractions. I think it is vitally important that students be proficient in working with fractions on all levels and with all types of computations. I have found in the past that students will know how to do a problem conceptually, but will struggle with simple computations, mostly involving fractions. Students need to be comfortable with these common computations so that they can start thinking of mathematics on a more difficult and conceptual level in higher classes.

The following is a list of websites that include games, interactives, practice items, and lessons that help students with multiplying and dividing fractions. Please feel free to add to this list as you explore this subject further. Also, please share those findings with the rest of us.

Lesson on Multiplying and Dividing Fractions

This website has two lessons. Lesson 1 involves multiplying fractions while lesso! n 2 works through dividing fractions. Both lessons have an ext! ensive l ist of practice problems with answers that students can use to check for understanding.

Math Workout: Multiplying and Dividing Fractions

These two websites are from the same source. These websites allow students to practice and work on dividing and multiplying fractions.

Multiplying and Dividing Fractions by Using Circles

These sites allow students to see the multiplication and division of fractions visually by using circles.

Dividing Fractions

This site allows students to enter their own fractions and then shows a detailed explanation of how the student’s problem was completed.

Math Playground: Multiplying Fractions

This website allows students to practice their skills on computer generated problems.

These are just a few of the activities that are available to students online. I hope that this is helpful in getting you started with your search. Next week we will move into another difficult subject, factoring.

Answers for dividing fraction

Some mathematics relating to the phi spiral

This is a more detailed look at one of the points made in my post on Nassim Haramein (number 3 on that page). I've written this in plain text for now, which I appreciate isn't easy to read. I'm aiming to set out the equations nicely sometime soon.


In the video "Phi Ratio Meets Vortex-Based Mathematics", Nassim Haramein can be seen arguing that the off-centre lo! cation of the emanation point of a phi spiral within an enclosing circle is connected to 9-fold 'resonances'. To illustrate this he divides a circle into 9ths and shows a W-shaped set of chords intersecting at a point that appears to be the emanation point of the phi spiral (illustrated above).

In fact it is straightforward to show that they're not related, and the two points are not the same at all.

1. True Phi Spiral

A phi spiral is a logarithmic spiral which increases in radius by a factor Ï•≈1.618 every quarter turn. The properties of logarithmic spirals can be derived with a bit of basic calculus. A full derivation would be a little too long to show in full here, and it turns out to be irrelevant to what was on Nassim's screen anyway, as he wasn't using a true phi spiral.

Briefly, t! hough... The first step is to calculate the radius of curvatur! e of the phi spiral, which is r√{1+(2 lnÏ• /Ï€)²} at a distance r from the emanation point. The second step is to work out the angle between a tangent of the spiral and the radius, which is arctan[(2 lnÏ• /Ï€)]. From these, with a little trigonometry, the distance of the phi spiral emanation point from the centre of its enclosing circle works out as √{1-1/[1+(2 lnÏ• /Ï€)²]} ≈ 0.29291 of the way along a radius from centre to circumference.

2. Intersection of Chords

[a diagram here of: an isosceles triangle with a vertex of 360/9 = 40º, and another with the same base but a lower vertex of 60º. The vertex of the first triangle is at the centre of the circle, for reasons of 9-fold symmetry; the vertex of the second is where the chords cross. What we're calculating is the distance between the two vertices. Wha! t can I use to draw this?]

The distance of the meeting point of the (W-shaped) chords from the centre can be calculated using basic trigonometry. The chords join points 1 & 5 and points 4 & 8 respectively of a regular 9-fold division of the circumference, as can be seen in the screenshot at the top of this page.

The result is {cos(Ï€/9) - √3 sin(Ï€/9)} ≈ 0.34730 of the way from centre to circumference. The 9-fold division of the circumference shown in Nassim's diagram, and which he specifies in the video, is clearly reflected in the formula.

This is nowhere near the emanation point of a phi spiral. The reason it appears close on the presentation is that the spiral used in the video is not a pure spiral at all, but an approximation.

3. Approximated Phi Spiral

Nassim's approximated phi spiral is m! ade from quarter-circles, each increasing in radius by a facto! r of Ï•. The two versions of the spiral are shown in green (pure spiral) and red (approximation using quarter-circles) on the diagram below. Where they overlap, it shows as yellow. They may look similar, but if you plot continuation circles around them they come out markedly different.

Image from wikipedia; creative commons license; modified from original.

This approximated spiral is constructed from circles embedded in squares, so we can use some basic geometry to calculate the position of the emanation point of this spiral – it lies √{1 - 2/√5} ≈ 0.32492 of the way from centre to circumference. To calculate this, note that the symmetry of the construction implies that it must lie at the point of intersection of the two diagonal blue lines in the diagram above. The eqations of the lines are shown on the diagram, in cartesian form, relative to an origin as shown on the diagram. They can be solved as standard simultaneous linear equations.

This value is close enough to the position of the intersection of the chords to look almost indistinguishable in the presentation. A careful viewer, however, will note that in Nassim's video, the lines do in! deed cross further from the centre than the emanation point, a! s can be seen in the still below.

A still from the video of Nassim's 'discovery'. In this close-up of an area below the centre of the circle, Nassim's diagonal chords are seen to cross below the emanation point of the spiral, not on it. I've added blue cross-hairs at roughly where the emanation point of this spiral would be.

The last two numbers (0.34730 and 0.32492, the ones relevant to Nassim's presentation) can easily be checked by careful construction of the shapes on paper or on a graphics package, and by measurement of the appropriate distances.

Conclusion

There is no mathematical relationship between a phi spiral and a circle split into 9ths. The relationship claimed in Nassim's presentation works only if you don't look too closely, and even then only if the spiral is approximated using quarter-circles.

For him to consequently make claims about the inspiralling of particles into a black hole is preposterous for a number of reasons. Firstly, inspiralling matter doesn't follow any logarithmic spiral (let alone a phi spiral). As matter spirals inwards it is diving in at an ever! steeper angle, even without resistance to motion. A logarithmic spiral, in contrast, approaches the centre at a constant angle. The resulting spirals are markledly different (see images below). Secondly, in the model he presents, as it is not a true spiral, the inspiralling matter would in fact have to follow a circular path for 90º and then, suddenly and discontinuously, follow a different circular path for another 90º, which is even sillier. And thirdly, even if it did, the values are still unrelated aside from their happening to be within about 1/44 of a radius of each other.

In addition, the 'interference patterns' that appear to reveal themselves when the spiral is replicated (in part 2 of this video) are purely an artifact of the approximation that he's used to build the spiral. If he had used a pure logarithmic spiral instead of quarter-circle ! building blocks, no such patterns could occur. On this topic, as with the others, we see his willingness to jump to vague but profound-sounding conclusions rather than investigate what he is thinking.

I suggest that it is clear from the above that the method of his arguments – if indeed there is one beyond spotting something that looks a little bit like something else – is very deeply flawed, very easily shown to be false, and lacking in both a basic understanding of mathematics and the imagination required to properly investigate straightforward mathematical situations. For this reason (indeed plenty more may be given, but this is reason enough), any of his claims to be employing mathematical or scientific methods to explain the underlying nature of the cosmos or anything else of any complexity should be seen as extremely dubious.
Above: spiral path of matter falling into a black hole (Using dr/dt∝1/r³; but assuming ω²∝r³. Can anyone help me out with a more accurate formula?)
Below: a phi spiral; clearly very different.

[Plotted using Wolfram Alpha ]

[back to main post]

mathematical construction

Euclid, Unique Prime Factorization and Sapir-Whorf

Unique prime factorization is a basic fact about the positive integers that many of us learn from a young age. Start with any positive integer and express it is as a product of prime numbers, say 105=3*5*7. This prime factorization is unique. That is to say, except for rearranging the order of these primes, there is no other way to express 105 as a product of prime numbers. Thus, for example, I don’t need to do any calculations to know that 105 is not equal to 11 *13. We can do this for any positive integer and regardless of what integer we start with we will get a unique result. (Another way of stating this is that if two people are both given the same number and told to factor it into primes, they will always get the same result as each other).

T! his idea is often called the "Fundamental Theorem of Arithmetic." Unique factorization is taught to children as young as 6th grade. When I have helped to teach number theory to high school students at PROMYS, many students take unique prime factorization for granted. Indeed, it often takes effort to convince them that the statement needs to be proved and is not just an obvious fact.

Despite the “obviousness” of unique prime factorization to the modern mind, the ancient Greeks did not know about unique prime factorization. This apparent failure of the ancient Greeks is well known to historians

of ma! thematics, but not known by many mathematicians. The only math! text I am aware of which mentions this is Hardy and Wright's “Introduction to the Theory of Numbers. Euclid in his seminal work “The Elements” has many results that are close to unique prime factorization. For example, Euclid was able to show that, if a is relatively prime to b and relatively prime to c, then a is relatively prime the produc of b and c. This result is almost as powerful as unique prime factorization, but it is not all the way there.

Why did the ancient Greeks not have this result? Simply put, they lacked the words to express the statement. To a mathematician of ancient Greece, there was no way to say "take a bunch of numbers and take their product! ." A product of two numbers x and y was easy; it was the area of a rectangle with sides of length x and y. Similarly, a product of three numbers -- x, y and z -- was the volume of a box with side lengths x,y and z. Fixed in this geometric mindset and with no concept of higher dimensions, the ancient Greeks could not talk about general products. Being unable to talk about general products, they apparently did not think about them.

To be sure, they groped towards this result. For example, Euclid proved a result that in modern language would be stated as follows: if n is the least positive integer divisible by some list of distinct primes p₁,p₂.p₃… p_k, then n is not divisibl! e by any prime not on the list. This result is trivial if one ! knows th at prime factorization is unique this is obvious since one can simply has n=p₁ * p₂... *p_k.

Apparently, the ancient Greeks, including Archimedes, Euclid and Diophantus, along with many lesser luminaries could not conceive of this idea because they lacked the language to express it. This failure is closely related to a hypothesis in psychology and linguistics known as the Sapir-Whorf hypothesis. Sapir-Whorf states that our language constrains our thought processes. There are different degrees of Sapir-Whorf. Very strong versions of the hypothesis are obviously false (if they were to be believed, translation between languages would be impossible) while very weak versions border on the trivial. This failure of the ancient Greeks is evidence for some strong form of Sapir-W! horf.

There's a lesson here for people other than psychologists. We cannot know how much we are missing simply because we lack the language to express an idea. Mathematicians over the last three centuries have taken this idea very seriously and spent much time trying to find optimal notation to express ideas. Yet, we cannot tell if there is some fundamental idea that we simply do not notice or appreciate because we lack the necessary language.

prime factorization