## Energy vs. Momentum – The Veritasium Video

A couple of days ago I noticed a blog post from Chad Orzel entitled How Deep Does Veritasium’s Bullet Go?.  It links to a video by Veritasium that asks which block goes higher, one shot with a bullet on center, or one shot with a bullet off-center.  Take a look at the video.

Before you go any further make an educated guess as to which block goes higher.  Rhett Allain at Dot Physics  goes through an analysis of what happens here.  Since Chad and Rhett did a pretty good job covering the physics I want to talk about this from a broader perspective.  Most of the answers I saw on different websites talked about energy conservation and only a couple actually mentioned momentum conservation.  I’m not completely sure why energy dominated the conversation, but I bet it has to do with the fact that more time is spent on energy than momentum in most physics curricula, and people seem more comfortable with energy, even though I think most students have a stronger conceptual grasp of momentum than of energy.

Momentum $\overrightarrow{p}$ is defined as the mass $m$ of an object multiplied by the velocity $\overrightarrow{v}$ of the object (for this post we’ll assume we are dealing with a classical system moving at much less than the speed of light) or $\overrightarrow{p} = m \overrightarrow{v}$  The arrow over the $p$ and $v$ mean that both of these are vectors.  A vector contains two pieces of information; how much, and which way.  A velocity vector tells you how fast the object moves and which way.  The momentum vector tells you how much momentum the object has and which way the momentum is pointing.  Energy, on the other hand, is a scalar quantity, which only provides information about how much, and has no directional component to it.  The kinetic energy $K$ of an object is defined as one half of the object’s mass $m$ times the magnitude of the velocity $v$ squared or $K = \frac{1}{2} mv^2$.  Notice than none of the terms in this equation are vectors (the magnitude of the velocity, or speed, is required, rather than the velocity, which is a vector).

The fact that energy is a scalar quantity makes it much easier to use than a vector equation.  If you’ve ever dealt with vectors you will remember that you need to keep track of three pieces of information, instead of the single number needed for a scalar.  The fact that there is no direction associated with energy which simplifies the math considerably.  However, to work with energy you have to throw out that piece of information that tells you “which way” which means you loose some idea about what your system will do.  Momentum requires you to keep track of more information, which means more calculations are needed.

Here is a simple example of the limitations of energy.  If I know the speed of a ball that is launched in the air I can use energy conservation to calculate the maximum height the ball could reach.  However, if the ball is launched at a $45^{\circ}$ this calculation won’t tell me how high the ball actually goes without a few extra calculations.  If I used momentum instead, I could figure out how high the ball goes, where it reaches its peak height, where it will land, and how long it was in the air.  Of course this will take a few more calculations to do, but all of the relevant information is retained.

The information loss yields another important (potential) limitation on energy.  Since energy does not conserve information about the direction of motion associated with energy, whenever two objects collide, like the bullet and block in this problem, the kinetic energy is able to split up into different directions.  In particular, collisions can cause the macroscopic kinetic energy of the incoming bullet to be transformed into internal energy of the system, either by heating things up or deforming the objects.  This means that it can look like energy seems to disappear altogether.  The energy is still there, it is conserved, but it is hidden from our eyes inside the block.

Momentum, with its directional nature, is much harder to hide (but not impossible*).  This means that however much upward momentum the system originally has (i.e. the bullet shot upwards) is exactly how much momentum we will be able to see after the collision (i.e. the block with the bullet embedded).  Since the bullet in both cases has the same initial upward momentum, both blocks after the collision will have the same final momentum, which means that must reach the same maximum height.

So what is the take-away message from all of this?  Energy, while useful, has its limitations.  You throw away useful information about the system when you start thinking in terms of energy.  Energy will help you figure out the limits of, say, how high the system might go, but it won’t always tell you exactly what will happen.  Momentum will give you a more complete view of what is really going on and is more likely to allow you to determine exactly how the system will move.  Be careful when using energy and make sure you are aware of the limitations of your tools.

* It is left as an exercise for the reader (or a future blog post) to figure out how momentum might seem to disappear from view
** It’s actually about the center of mass of the wood block and bullet system, but given the low mass of the bullet these two points are pretty close together.
Posted in Learning Physics | Tagged , | 4 Comments

## Why is Multiplication Harder than Addition?

I’m in the process of writing a couple of blog posts about slide rules, and an interesting questions occurred to me, which spawned this post. Why is addition so much easier than multiplication? The power of slide rules comes from the fact that multiplying two numbers together is equivalent to adding two logarithms. The slide rules made life easier for scientists, engineers, and financiers for centuries by reducing multiplication problems down to simple addition.

I decided to figure out how many operations adding and multiplying takes. The model I’m using is based on Feynman’s model of a simple computer from his book Feynman Lectures on Computation. Imagine that you have a person who doesn’t know how to add or multiply numbers, but they are very organized and very good at following directions. They have a sheet of paper in front of them with the problem they are trying to solve (either adding or multiplying numbers) and the only tool they have is an addition table which allows them to look up the answer to x+y for all numbers between 0 and 9. Additionally, they have a multiplication table for all numbers between 0 and 9. Every time they have to access the additional table or multiplication table it is considered a single operation. The question I want to know is, in general, how many operations does it take to add or multiply a series of numbers together? I’m sure one of my mathematician friends could rattle the answer off the top of their head, but I wanted to see if I could figure it out on my own.

Let’s start simple, with two 2 digit numbers (to hopefully avoid confusion I’ll use numerals to denote the number of digits and words to denote the number of numbers).

$\begin{array}{@{}cr} & 31 \\ + & 17 \\ \cline{2-2} & \\ \end{array}$

The computer (note that the word computer used to refer to a person who calculates numbers) pulls out his chart to look up $1 + 7 = 8$ and writes 8 down in the first column.
$\begin{array}{@{}cr} & 31 \\ + & 17 \\ \cline{2-2} & \ 8 \end{array}$
Next he looks up $3 + 1 = 4$ and writes that down.
$\begin{array}{@{}cr} & 31 \\ + & 17 \\ \cline{2-2} & 48 \end{array}$

Since the computer used the look-up tables twice, it took two operations to add two 2 digits numbers. Let’s extrapolate to adding more numbers. For simplicity, I’ll assume that all numbers that are being added have the same number of digits. If you add three 2 digit numbers together it ends up taking a total of $3\times 2 = 6$ operations. Remember that your addition tables only allow you to add two digits together at a time so you will need to add the first two numbers together first, then add the third number to this results.
$\begin{array}{@{}cr} & 31 \\ + & 17 \\ + & 12 \\ \cline{2-2} & \end{array}$
The first two numbers give you 48 and took four operations, and adding 12 to 48 takes another two operations, for a total of six.

We can extrapolate to see that adding M 2 digit numbers together takes a total of 2M operations. You can also easily see that if you expand this even further to consider numbers with N digits, it will take $N\times M$ operations to add those numbers together. This is actually a lower bound on the number of operations, because frequently you will add two numbers and get something larger than 10, which requires you to add a carry digit to the next column over. It turns out that, on average, adding M numbers with N digits, it will take $\left( M + \frac{1}{2} \right) N$ operations.

## Multiplication

Does multiplication take more operations to complete? Let’s take a look.
$\begin{array}{@{}cr} & 13 \\ \times & 21 \\ \cline{2-2} & \end{array}$

Our computer starts off by looking up $1\times3 = 3$ and then $1\times1 =1$. The next step is then looking up $2\times3 = 6$ and $2\times 1 = 2$ so we’ve completed four operations and have the following results:
$\begin{array}{@{}cr} & 13 \\ \times & 21 \\ \cline{2-2} & 13 \\ & 260 \\ \end{array}$

Now we need to add the two numbers together. The number of operations is either one or three, depending on whether you count addition by zero as an operation. I’m going to count it as an operation since we assume our computer doesn’t understand anything about numbers and needs to look up even this simple process. This means that multiplying two 2 digit numbers together takes seven operations, which is clearly larger than the four operations taken to add two 2 digit numbers together. To come up with an equation for multiplying more numbers with more digits we need to consider that adding another digit means you now have to multiply that one digit by all of the other digits in the other number. Multiplying two 2 digit numbers yields $2^2 = 4$ multiplication operations so multiplying two N digit numbers will yield $N^2$ operations. Once you are done multiplying the digits together, you have two numbers with 2N digits that you need to add together so these combining two N digit numbers takes roughly $N^2 + 4N$ operations. Note that for N = 2 the equations says that it should take 8 operations but I only counted 7 in my example. This is because I assumed two 2 digit numbers yields a four digit product, which only occurs if all of the digits are larger than 3.

Things start to get a little tricky when I try to extend this to multiple numbers so I’m going to leave that for another day. The end message is that the number of operations to multiply numbers together scales as $N^2$ while addition scales as $N$. As the number of digits increases, multiplication rapidly starts taking more individual operations to get a results. This means that if you can somehow simplify a multiplication problem so it only requires additions, you might be able to save some time and effort. Stay tuned for my upcoming posts on slide rules.

Posted in Just for Fun | | 2 Comments

## Students Are Not Customers

The latest trend among college administrators is to say we need to be treating our students like customers. The idea is that educators need to improve “customer service” in order to give students a better educational experience. This model doesn’t make any sense. It almost feels like administrators have taken a few MBA courses and latched onto this idea of students-as-customers as the solution to the problems faced by higher ed. With the latest financial crisis resulting in reduced endowments, the trend of states cutting funding to higher education, and a soon to be shrinking pool of students, universities are looking for ways to keep enrollment up, but treating our students as customers is not the way.

What is the biggest problem with the students-as-customers model? This leads to a sense that professors somehow owe students a passing grade and the university owes the students a diploma. This sense of entitlement is psychologically damaging to the faculty. I know this isn’t the mindset most administrators start out with, but it is an unintended consequence of the students-as-customers idea. We already deal with students (and even parents) who feel entitled to good grades in our courses (“My special little snowflake, Jonny, should get an A because he is going to med school”). I think the business model further reinforces this model and can make faculty feel like they aren’t supported by administration.

Let’s stick to the overall business mindset for a moment and take another look at universities. First off, students are not customers, nor are they products. Graduates are what higher ed “produces” and it is employers and graduate schools that “purchase” the products. This makes students the raw materials that have the potential to be turned into finely crafted diploma recipients. To keep employers happy, we need to be turning out high quality students. Following this idea, educators should be culling weaker students and admitting only the highest caliber of applicants. The problem with this model is that universities (especially public universities) have a larger responsibility to society to provide educational opportunities to residents. We have a responsibility to provide students the best chance of success we reasonably can. The business model for universities fails because we have responsibilities to differing groups that sometimes conflict. What employers and grad schools need is different than what society needs, which is different than what individual students need.

Posted in Just for Fun | Tagged | 3 Comments

## How Many Objectives Should I Have?

One question I frequently see people asking about Standards Based Grading is how many standards or learning objectives are reasonable for a class. I came up with a back of the envelope calculation to give you and idea what the maximum number of learning objectives your course should have.

${Learning\ Objectives} = \frac{\left(Assessments\ per\ week\right)\left(Objectives\ per\ assessment\right)\left(Weeks\ per\ semseter\right)}{\pi\left(Proficiencies\ per\ objective\right)}$

Most of the terms in the equation* are self-explanatory, but “proficiencies per objective” is how many times you expect a student to demonstrate proficiency in order to complete a particular learning objective. The factor of pi is thrown in because (1) without that factor the equation gives a number that is too large and (2) every good scientific equation needs a factor of pi in it somewhere.

* The observant reader will note that the units of the above equation don’t work out. This is because “Objectives per assessment” should really be “proficiencies per assessment”

## LOBA and Standards Based Grading

I should probably clarify the distinction between Learning Objectives Based Assessment (LOBA) and Standards Based Grading (SBG). LOBA is a particular implementation of SBG. I want to be clear that LOBA isn’t some brand new grading paradigm that I created out of the blue. It is my take on what many others have done in implementing SBG in their classrooms.

So the next question you will ask is “what is SBG?” The heart of SBG is (1) to get students focused on concepts and skills rather than points, (2) make sure grades are an accurate measure of student understanding, and (3) give students multiple chances to master material. This is essentially the same as the LOBA philosophy. The distinction between LOBA and SBG is really in the mechanics. There is no one correct way to implement SBG and different instructors have tried a wide variety of techniques in terms of determining final grades, reassessing, the number of standards, and so forth. In LOBA, the final grade is determined by the fraction of learning objectives a student completes. There are typically A-level and C-level learning objectives (and sometimes more), and completion of most lower level learning objectives is required to earn higher grades. The number of learning objectives tends to be higher than some SBG implementations because LOBA focuses on more discrete skills and concepts, rather than big-picture standards. Reassessment opportunities have limitations to make the workload more manageable. I know there are SBG-ers out there using many and sometimes all of these features so it is fair to say that LOBA is a subset of SBG.

When I started using SBG I ran in to a number of faculty who would ask if SBG had anything to do with the state education standards, usually with a pinched face you’d make when finding six-month-old leftovers in the back of your fridge. University faculty tend to be dismissive of, unhappy with, or skeptical about the state-mandated education standards. Learning objectives are something that every educator is aware of and it doesn’t have the same stigma. I initially called my implementation Learning Objectives Based Grading (LOBG) but several of my friends made fun of how “LOB-G” sounded (I know peer pressure is a silly reason to change, but I wanted to be taken seriously. Sigh).

A lesser reason for using a different name than SBG was I didn’t feel right speaking for the whole SBG community. I was new to this type of grading paradigm and I felt better talking about my particular implementation rather than seeming to speak for all SBG-ers. I realize that it might limit my reach, since people will be searching the internet for “SBG” and not “LOBA”, but hopefully people will eventually start to equate LOBA with a type of SBG. It may turn out that not using the SBG moniker is a mistake and I may have to start referring to what I do as SBG at some time in the future.

Incidentally, according to the literature1, what we have been calling standards-based grading is actually standards-referenced grading. In true standards-based grading, students do not progress to new material until they have successfully demonstrated proficiency on a particular standard. In standards-referenced grading, students performance is reported (or referenced) to the standards but students are allowed to proceed to the next level.

1 – Marzano, R. J. (2010). Formative assessment & standards-based grading. Solution Tree, p18.

## Light Doesn’t Stop – “Stopped Light and Image Storage by Electromagnetically Induced Transparency up to the Regime of One Minute”

I still remember the group meeting in grad school when someone brought up a news article stating that a researcher had slowed light down to the speed of a bicycle. We got a good laugh at the inaccuracies and clear misunderstandings in the news article. It was a little disturbing, then, to see similar articles pop up all over the web. The main problem with the articles was they claimed that scientists were slowing light down, when in fact they were slowing light pulses down. What’s the difference, you ask? Stick around and I’ll show you. When I recently saw a news article claiming that scientists had brought light to a complete stop, I felt the veins in my head start to throb again, and I couldn’t let it go without saying something. Hence this post. So what did they actually do$^1$? They were able to store the quantum properties of a light pulse for up to a second and then retrieve the information in a new pulse that was identical to the original. Storing the pulse and how then maintained the integrity of the information was the exciting aspect of this experiment. How do you go about storing the information from a pulse of light? You first need to compress the pulse so it is small enough to fit inside your apparatus. Since light moves so fast ($3\times10^8 \frac{m}{s}$), a pulse that is 1\micro s will be 1 km long! It is rather difficult fitting that into a lab room, let along a small crystal used for storage. The way you spacially compress the pulse is to slow it down as it enters the medium (in this case a silicate crystal of yttrium doped with Praseodymium (I’m not even sure where that is on the periodic table). Using something called electromagnetically induced transparency (EIT) the group velocity of light slows down, so that as the light pulse hits the crystal it is just like traffic hitting construction, the kilometer long pulse piles up into a centimeter long pulse. It is important to note that it is the group velocity that slows down to around 1 km/s and not the phase velocity. What is the difference? I’m glad you asked because this is at the heart of my complaint about the claims of “slowing light down to the speed of a bicycle”.

## Phase Velocity and Group Velocity

Classically speaking, light is an electromagnetic wave which means that the magnitude of the electric and magnetic fields vary sinusoidally. The phase velocity is how fast a particular peak or valley of the light wave moves in space.

Figure showing group velocity and phase velocity. Red dots move with phase velocity (following a peak of the wave) and green dots move with group velocity (following the nodes between pulses)

The red dot in the image above is moving with the phase velocity of the waves. However, to have a wave pulse, you must have many different waves of different frequencies. When you combine multiple frequencies of a wave you end up with interference between the waves that results in a phenomena called beating, which yields a slowly varying envelope superimposed on a rapidly varying wave. In the figure above you can see that the green dots are in between each of the larger envelope pulses and that the green dots are traveling slower than the red dots (notice how the red dots catch up to the green dots and pass them). This happens because, in some media, the phase velocity of light depends on the frequency. By introducing a very slight difference in phase velocity can result in a huge change in the group velocity. The light pulses in the experiment are the envelop you see in the figure and move at the group velocity. So the key here is to somehow have a medium where the phase velocity changes as the light frequency changes, so the pulse group velocity can be very slow. This is where electromagnetically induced transparency (EIT) comes in.

## EIT

Atomic level configurations for electromagnetically induced transparency. I will only talk about the Lambda configuration

This image shows a couple of different energy level configurations but I’ll focus on the lambda configuration (so named because it is shaped like the Greek letter lambda $\Lambda$). An energy level is possible state that an electron bonded to an atom can occupy. Electrons that are orbiting closer to the nucleus of the atom have lower energies than electrons that have orbitals that are farther away. On this diagram, states with higher energies are above states with lower energies, and the vertical distance between states represents the energy it would take to go from one state to the other (Don’t worry about what the horizontal displacement means, all we care about now is the vertical arrangement). If an atom is in one energy level, it can be encouraged to transition to a different energy level by absorbing or emitting light. The experimenters used laser light to encourage transitions between differing energy levels. The medium you shine your light through (in this case the silicate crystal of Yttrium) consists of millions and millions of atoms with similar energy level configurations. This means that if you shine a light pulse through that has an energy (which depends on frequency) that corresponds to any of the transitions between two levels (say between the |1> and |3> states), the light pulse is going to get absorbed by the atoms and not make its way through. However, if you shine a strong control laser (shown in red in the figure and with frequency $\omega_c$) and a weaker probe beam (shown by the black arrow and denoted to have frequency $\omega_p$), and you tweak the lasers just right, you can get interference between the states that prevents light from getting absorbed on the transition between states |1> and |3> (let’s call that frequency $\omega_x$). This is EIT; the medium is now transparent to light with an energy corresponding to $\omega_x$ because that light is no longer absorbed. One interesting result of EIT is that right around frequency $\omega_x$ is that the phase velocity of light varies significantly with frequency.

Blue curve is index of refraction vs. frequency. Grey curve is absorption of light vs frequency

The blue line corresponds to the phase velocity (actually the index of refraction) and the horizontal axis is frequency. The grey curve shows the absorption of light so you can see the transparent window (the dip in the curve) where the light is not longer absorbed. Since the phase velocity varies, any pulse of light with frequency near the transparent window will have a very slow group velocity. By tweaking the properties of the system you are able to get light going 1 km/s in this experiment (in other experiments involved cold gases, researchers have achieved speeds of 17 m/s in an ultracold atomic gas$^2$).

## Storing the Pulse

Reading some of the popular articles, they make it sound like they actually slow the light down enough to stop it, but that isn’t how it works. While the pulse is moving much slower than it did in air, it is by no means standing still. What you need to do is get the medium to absorb the photons, but in a way that preserves the quantum properties of the light. This is done by switching off the control laser beam, which means that the medium is now able to absorb light with energies corresponding to the |1> to |3> transition. As the control beam is turned off, the transparent transition can now absorb light, and the quantum properties of the light are stored as spin excitation of the atoms. The important point here is that the energy from the electromagnetic field making up the light pulse is now transferred to the atoms, that the light gets absorbed, and there is no more light pulse. Now comes the hard part, making sure that the stored information is not lost, and this is where this experiment shines. By applying various magnetic fields and applying a microwave pulse, the experimenters were able to keep the atoms isolated from the effects that cause decoherence (basically, things get bumped and jiggled enough that the information is lost). What is even more amazing is that they used an adaptive algorithm to tweak the parameters of the magnetic fields and microwave pulse to maximize the storage lifetime. They did such a good job that they were able to get close to the theoretical maximum storage time for this particular type of crystal, but this technique can now be applied to other materials that might have longer possible storage times.

## So why is this experiment exciting?

Overall, the experiment is about storing and retrieving the quantum mechanical properties of a light pulse. This has applications to quantum computing and quantum cryptography, which rely on quantum entanglement. The big deal here is that they managed to store light pulses in a solid-state device, a doped silicate crystal. Most of the previous storage experiments had focused on using cold gases, but a lot of people feel that solid-state devices will be needed to scale up to larger projects involving hundreds or even thousands of light pulses. Solid-state devices are more difficult because of the strong interaction between atoms of the crystal, which makes it harder to isolate the atoms that are used to store the pulse. 1 Georg Heinze, Christian Hubrich, and Thomas Halfmann (2013). Stopped Light and Image Storage by Electromagnetically Induced Transparency up to the Regime of One Minute Physical Review LEtters DOI: 10.1103/PhysRevLett.111.033601 Note: The article is behind a paywall, but you can read a summary at http://physics.aps.org/articles/v6/80?from_TRM_site=Yttrium 2 Hau, L. V., Harris, S. E., Dutton, Z., & Behroozi, C. H. (1999). Light speed reduction to 17 metres per second in an ultracold atomic gas. Nature, 397(6720), 594-598.  http://www.nature.com/nature/journal/v397/n6720/abs/397594a0.html Sorry, this one is behind a paywall too.

Edit: I found a nice article by Chad Orzel talking about slowing light.  Check it out:  http://scienceblogs.com/principles/2010/01/05/controlling-light-with-light/

Posted in Teaching Physics | | 2 Comments

I was looking back at my previous posts and noticed I hadn’t talked explicitly about how I keep track of grades for assessments, only about how I use Python and Excel to make my life easier.  I used to include learning objective numbers along with each problem so I could write scores on the sheet but I found that was a problem because (1) I’d have to go back through each assessment and copy the scores to my grade book and (2) students would memorize which skill corresponded with each learning objective which made it much easier for them to solve problems.

Learning objectives listed next to the assessment problems.

Front page of student grade sheet (top half of Excel worksheet)

Back of student grade sheet (bottom half of Excel worksheet)

The benefit of this system is that I’ve reduced the number of grading errors to almost zero, that I now have a record of what each student got on each assessment, and that I have a list of the comments I made to the students to reference later.

My only concern with this setup is that having the scores on a separate page from the problem may be an additional barrier to students.  Based on anecdotal evidence this doesn’t seem to be an issue (the second I hand back assessments I see students flipping through the pages to see which problems they got proficient marks for) but I must remind myself that the plural of “anecdote” is not “data”.

The other component to my grading system is using Python along with Excel to copy all of the scores from the student worksheets to a master Excel spreadsheet (see the previously mentioned post on using Excel and Python together).  If anyone needs help setting this up I am more than happy to help troubleshoot.

I’ve attached a copy of a blank assessment workbook, which you are more than welcome to copy and use.  Let me know if you try it out:  Sample Blank Assessment Gradebook