Blog of Andrés Aravena

# Keep your mind sharp on August

23 August 2018

Why do rich kids learn more than poor kids? According to the book “Outliers”, the “achievement gap” between kids of high-income and low-income families is not that rich kids learn more, but that they forget less in the summer holiday. The data, taken from a well researched paper 1 published in 2001, shows that poor kids learn at a similar rate than rich kids, and even sometimes faster (on average). The difference is on what they forget during summer holidays.2

The book “Outliers”, by the Canadian journalist Malcom Gladwell, is indeed one of my favorite books, since it opened my mind to different ideas. There is a translation into Turkish language, although if you are reading this then you probably will enjoy the original one. Keep reading to learn how you can get it for free.

Following this idea, I proposed to some students a couple of challenges to keep their mind active during the holidays. Let’s consider the following scenarios

1. You are going to prepare tea. The water is in a 20 liter bottle with an air pump that you press to get water out of a plastic pipe. When the bottle is full of water, you press and the water gets out immediately. But when the bottle is half empty you need to press several times and the water comes out a few seconds later. Hey, you even drop some water, because you pressed too much.

2. Now you put the water into the tea maker and you boil it. You look at the cup from the top when you serve tea. You pour tea and the cup gets darker and darker. Then you put hot water on the tea and the cups becomes less and less dark.

3. Bonus question: Why you need to boil water to prepare tea?

The question in all three cases are:

• why does this happen?
• How can you explain this?
• Is there some mathematical rule?
• What experiment can you do to find such rule?
• What simulation can you make to predict the effects of such rule?
• How can you test and corroborate your explanation?

I challenge you to think about it and write me some words before September. The best answer gets a book as a gift, either “Outliers” or a good science fiction one. I’m not asking for complete solutions, just for ideas. Any idea is valid, I will give you more ideas on the next weekend.

It took me more than one weekend to write this followup, but so far nobody complained. The only person that has accepted the challenge so far is Eda Şamiloğlu. These are her answers (with minor typo corrections) and my comments:

1. When you put something into the glass of water, the total volume rises up. That something’s volume, exchange equal volume of water.

In this case of the question, we press the air into the water and it exchange water equal volume of air. If the bottle is full of water will be boil over. If not, water will rise up of course with a equal volume of air. But with normal teapot, water do not boil over, why? Because the pump of teapot has some pore3 and air go out again without exchange water volume.

These arguments are correct, but there are missing parts. When we put extra air in the bottle several things can happen. It can increase the volume, as Eda says, or increase the pressure, like in a compressed air tank. In some cases it can even reduce the temperature, as it happens in the freezer. The essential formula is $PV=nRT$ where $$n$$ is the number of molecules, $$R$$ is Boltzmann’s constant, and $$P, V$$ and $$T$$ are obviously pressure, volume and temperature. This is known as the law of gases4. When we use the air pump we don’t observe any noticeable change of temperature, so the extra air essentially increases pressure, at least in the short term5.

My theory is that each time we push the pump we add more or less the same number of air molecules, so $$n$$ increases and $$P$$ also. The pressure of air pushes the water with a force $$F=PA$$, where $$A$$ is the area of the water surface. That force makes water move, with acceleration equal to $$F/m,$$ where $$m$$ is the mass of the water, which depends on the volume and density of water.

Why there is a longer delay when the bottle is half empty? Because when the volume of air is half a bottle, then the extra air only increases the number of molecules (and thus the pressure) only a little, so the force is small. Sometimes we have to press the pump several times to get some water. On the contrary, when the bottle is full of water, every punch of the pump increases the number of atoms by a lot so we get more pressure and acceleration.

How can we test this theory? I think we can simulate the system in the computer and predict how long it will take for the water to get speed depending on the volume of air. Maybe we can use the Bernoulli’s formula we saw in class. The simulation can give us predictions that we have to test with some experiments.

The new challenges are:

• To write the simulation and run it
• To design an experiment
• To do the experiment (optional)
1. If we think that tea is homogenous mixture (it is not, but it is easy to think), it is mean every ml of water has equal number of black points. When you pour tea, cup gets darker and darker because the black points accumulate and don not allow the light to pass. In result we see the cup darker and darker. When we put hot water, cups became less and less dark, because we change the black point number of per 1 ml water. It is called “concentration” in chemistry. Attention: pouring more tea does not change the concentration but putting water does change it. (concentration=mol/volume)

Seems good to me. The key idea is that optical density depends on tea concentration. Then the opacity depends on the dimension of the liquid. If we look from the side the tea keeps the same color during the first phase, and gets clearer in the second phase. If we look from the top, the tea gets darker at the beginning, and then it it should keep the color in the second phase.

We can write a formula for the tea concentration, something like $f(V)=\begin{cases}C\text{ if }V<V_0\\C/V\text{ if }V\geq V_0\end{cases}$ and then we can calculate what will be the color of the tea cup given its dimensions. We can simulate this on the computer.

Now we need to decide an experiment. I imagine for example recording a video of a glass from the top and another from the side, while we pour tea and water. Or better, we put a mirror at 45 degrees on the side of the glass, and we record only one video containing vertical and lateral view of the glass.

If we are clever, we can find some library in Python or R (or whatever) that can read the video. Then we can write a program that measures the color on the center of each glass and produce a vector of darkness v/s volume for each glass. If the data does not match the model, then the model has to be adapted.

Can you propose an easier experiment?

1. Heat increases the dissolution. Remember this basic example: in warm water, sugar melt quicker than in cold water.

Yeah, they say so, but tea is not sugar. And I would say temperature instead of heat, since a bigger bottle at the same temperature has more heat. I propose this experiment:

• Let’s put a bottle of water in the fridge and wait until it is cold but not frozen
• Boil half liter of water
• Fill a glass with half cold and half hot water. Let’s call this the 50% glass. Put a label on it
• If we have enough glasses, we fill one with half of 50% and half cold (call it 25%) and another with half of 50% and half of hot water (choosing a suitable name).
• Obviously, we also have a 0% and a 100% glass.
• If we have a thermometer, we can even measure the temperature of each glass, but this is not strictly necessary.
• Then we put all the glasses in front of a white wall with good light.
• We record the experiment with a slow-motion camera. Many cellphones can take video in Slo-Mo.
• We put a tea bag on each glass without shaking it (that is for another experiment).
• We need to record for several minutes. I guess 10 minutes would be enough, but we can try longer periods.
• If we are clever, we can find some library in Python or R (or whatever) that can read the video.
• Then we can write a program that measures the color on the center of each glass and produce a vector of darkness v/s time for each glass
• Finally, we have to find a formula that fits the darkness depending on time and water mix.

If Eda is right, we will see that the speed of tea dissolution increases with temperature. And we will know how much increases.

## Remaining questions

• How would you do it using sugar instead of tea?

• Do you have any other idea (not necessarily original or better) on how to do this?

• What are the implicit hypothesis that we use here?

1. Karl L. Alexander, Doris R. Entwisle, and Linda S. Olson’s study on the impact of summer vacation is called “Schools, Achievement, and Inequality: A Seasonal Perspective,” published in Education Evaluation and Policy Analysis 23, no. 2 (Summer 2001): 171-191.↩︎

2. I could not find that paper, but instead I found a more recent one that tells the same story.↩︎

3. valve?↩︎

4. I confess that this is the only thing I learned in my Chemistry course when I was student, despite having one of the most famous professors of my faculty. He used to speak in a very monotone voice and I felt asleep on each class.↩︎

5. Notice that water and solids are mostly incompressible, so pressure does not change very much, only volume. But air is a compressible gas. That is why we put it inside car wheels.↩︎

Originally published at https://anaraven.bitbucket.io/blog/2018/keep-your-mind-sharp-on-august.html