Why do not the sizes of Venus and Mars as viewed from Earth change during the course of the year?

Just before his death, in 1543, Nicolaus Copernicus published in his book On the Revolutions of the Celestial Spheres a Heliocentric model of the universe, that is, a model of the universe that placed the Sun rather than the Earth at the center of the universe.  This is considered a major event in the history of science, triggering the Copernican Revolution and making an important contribution to the Scientific Revolution.

Geoz wb en.svg

According to Copernicus' model, since the Earth circulates the Sun in an orbit outside that of Venus and inside that of Mars, the apparent size of both Venus and Mars should change appreciably during the course of the year. This is because when the Earth is around the same side of the sun as one of those planets it is relatively close to it, whereas when it is on the opposite side of the sun to one of them it is relatively distant from it. When the matter is considered quantitatively, as it can be within Copernicus's own version of his theory, the effect is a sizeable one, with a predicted change in apparent diameter by a factor of about eight in the case of Mars and about six in the case of Venus.

On the other hand, according to the Ptolemaic system (the Geocentric model) Venus and Mars should not change appreciably during the course of the year because its epicyclical motion implies only a small change in distance from the Earth.

However, when the planets are observed carefully with the naked eye, no change in size can be detected for Venus, and Mars changes in size by no more than a factor of two. This gives us strong evidence for the Geocentric model and refutes the Heliocentric model! How is this possible?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

Something happens with the light

If we split a collimated light beam by using a half-silvered mirror, then the two resulting beams (A and B) have exactly the same intensity. Since the light is made of photons, that means that half of the photons go through path A, and the other half through path B.
If we now reflect both beams by a mirror and the two beams then pass a second half-silvered mirror and enter two detectors as explained in the picture:
then we expect the A beam to be split into two beams. We will call them A1 and A2. A1 goes to dectector 1, while A2 goes to detector 2. Each one contains 50% of A-photons, that is, 25% of the photons of the original light beam:
On the other hand, we also expect the B beam to be split into two beams. We will call them B1 and B2. B1 goes to dectector 1, while B2 goes to detector 2. Each one contains 50% of B-photons, that is, 25% of the photons of the original light beam:
So the amount of photons that should arrive to detector 1 is 25% + 25% = 50%, and the same for detector 2:
Nevertheless, once we have carried out the experiment, what we found is that 100% of photons arrive to detector 2  and no photon arrives to detector 2!
Moreover, what is even more puzzling, if we obstruct channel A (or B, it does not matter), then we detect the same number of photons in detector 2 as the number detected in detector 1. Are you able to figure it out? Try it!

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

Why do ice cubes melt faster in fresh water than in salt water?

The melting point of a solid is the temperature at which it changes state from solid to liquid at atmospheric pressure. When considered as the temperature of the reverse change from liquid to solid, it is referred to as the freezing point.

The freezing point of a solvent is depressed when another compound is added, meaning that a solution has a lower freezing point than a pure solvent. This phenomenon is used in technical applications to avoid freezing, for instance by adding salt or ethylene glycol to water. If you live in a place that has lots of snow and ice in the winter, then you have probably seen the highway department spreading salt on the road to melt the ice.

Now, let us consider the following experiment:
  1. Make two almost identical ice cubes.
  2. Mix 1 teaspoon of salt in an 8 oz. cup of water. This will be our salt water cup.
  3. Fill a 8 oz. cup with water, but with no salt added. This will be our fresh water cup
  4. Place one ice cube into each cup simultaneously. Which ice cube do you predict would melt the fastest?

Naively, one would think that, according to the previous information, since salt lowers the freezing/melting point of water, the ice cube in the salt water cup should melt the fastest.

Nevertheless, if you carry out the experiment, it leaves no doubt. The ice cube in the fresh water cup melts faster!

Why do ice cubes melt slower in salt water? Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

I will give you a clue: repeat the experiment, but this time, after you place the ice cubes in the cups, wait 30 seconds and add a couple of drops of food coloring to each cup without disturbing the water in the cups.

Why does the death of a living being affect the decay of carbon-14?

Carbon-14 is a radioactive isotope of carbon with an atomic nucleus containing 6 protons and 8 neutrons. Carbon-14 decays into nitrogen-14 through beta decay:
By emitting a beta particle (an electron, e-) and an electron antineutrino (νe), one of the neutrons in the carbon-14 nucleus changes to a proton and the carbon-14 nucleus becomes the stable (non-radioactive) isotope nitrogen-14.
The equation governing the decay of a radioactive isotope is
$$ N=N_0 e^{-\frac{t}{\tau}}$$
where No is the number of atoms of the isotope in the original sample (at time t = 0, when the organism from which the sample was taken died), and N is the number of atoms left after time t. On the other hand, the mean-life τ is the average or expected time a given atom will survive before undergoing radioactive decay.
Since the amount of carbon-14 inside a piece of wood or a fragment of bone decrease as the carbon-14 undergoes radioactive decay, measuring the amount of carbon-14 in a sample provides information that can be used to calculate when the animal or plant died. The mean-life of carbon-14 is 8267 years, so the equation above can be rewritten as:
Nevertheless, radioactive decay is a process that takes place inside the nucleus, so nor a change of temperature neither chemical reactions affect radioactive decay. Carbon-14 atoms inside a living being are decaying after and before the living being dies. So why is this method used efficiently to measure when the living being died? How do we know No, the amount of carbon-14 the living being had at the moment it died, if carbon-14 was also decaying when the plant or the animal was alive?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

Does the expansion of space apply in the solar system?

The prevailing cosmological model for the universe accounts for the fact that the universe expanded from a very high density and high temperature state, and that nowadays the expansion is even accelerating. This is an expansion of space, that is, the increase of the distance between two distant parts of the universe with time. It is an intrinsic expansion whereby the scale of space itself changes. This is different from other examples of expansions and explosions in that, as far as observations can ascertain, it is a property of the entirety of the universe rather than a phenomenon that can be contained and observed from the outside.

By NASA/WMAP Science Team - Original version: NASA; modified by Ryan Kaldari, Public Domain, https://commons.wikimedia.org/w/index.php?curid=11885244

Since it is an intrinsic expansion, it is natural to think that the planets in our solar system are expanding with time, as universe is. Moreover, our measurement devices should be expanding too. But, taking into account that a measurement is the assignment of a number to a characteristic of an object by comparing with other objects, why were we able to measure the expansion of the universe if our devices are expanding too?

Does the expansion of space apply to the objects inside our solar system?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

Why are cosmic ray muons decaying more slowly than predicted?

Muons are unstable elementary particles. They are heavier than electrons and neutrinos but lighter than all other matter particles. They decay via the weak interaction. A muon decays most commonly to an electron, an electron antineutrino, and a muon neutrino:

The mean lifetime, τ = 1/Γ, of the muon is (2.1969811±0.0000022 ) µs. That means that every 2.19698 µs the population of muons is reduced by a factor e=2.71828.
An experiment compared the population of cosmic-ray-produced muons at the top of a mountain, whose height is 2 km, to that observed at sea level. Those muons were traveling at 0.95c, where c is the speed of light, so they arrive to the sea level t=7 µs later. At the top of the mountain the measured population was No=563 muons per hour. Therefore, according to the decay law, the expected population of muons at the sea level should be:
$$ N=N_0 e^{-\frac{t}{\tau}}=23$$
muons. Nevertheless, 413 muons where measured, so the muon sample at the sea level was only moderately reduced! The muons were decaying about 10 times slower!
Are you able to explain this anomaly? Try it!

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

Why did the hunter wound the bear?

A hunter is located 100 meters due South from a bear. Then he goes East 100 meters. After that, he looks to the North and he shoots to the North, wounding the bear. Nevertheless, we know that the bear did not move!

You will be even more puzzled after realizing that the question is:

What color is the bear?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

The Moon is getting further away from Earth. Where does this extra energy come from?

All bounded orbits where the gravity of a central body dominates are elliptical in nature. In the case of the Moon orbiting the Earth, the eccentricity of the ellipse is so small (0.055) that it is almost a circle:
Therefore, the gravitational force Fg that the Earth exerts on the Moon is perpendicular to Moon's velocity v, so it is a centripetal force Fc, making the trajectory of the Moon bend:
$$ F_{g}=F_{c} \\
\frac{GMm}{r^2}=\frac{mv^2}{r} $$ where G is Newton's constant, M is Earth's mass, m is Moon's mass and r is the radius of the orbit.

This implies that the kinetic energy of the Moon is
$$ which is smaller than the absolute value of the potential energy
$$ So the mechanical energy of the Moon is

We know that at the time of its formation, the Moon sat much closer to the Earth, a mere 22,500 km away, compared with the 402,336 km between the Earth and the Moon today. So the Moon is getting further away from Earth, now at the rate of 3.78 cm per year. Nevertheless, according to the last equation, a larger r means that the Moon has more energy every year. Is its energy non conserved? Who is giving energy to the Moon?

Can an object exceed the speed of light if we push it for enough time?

The second Newton's law of motion establishes that, in an inertial reference frame, the vector sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration vector a of the object. That means that if we apply a constant force on a body without friction, then the object will move with constant acceleration, increasing its speed by the same amount every second.
The acceleration is proportional to the force, so if the net force is 100 Newtons and the mass is 2 kilograms, the acceleration will be 50 meters per second every second. But if the force is 2 N, the body will increase its speed by 1 m/s every second. Notice that this is not a huge acceleration. Nevertheless, if we keep pushing and wait for 300000000 seconds (9.5 years) the object will move faster that light.

But we know that nothing can exceed the speed of light. This is a well-established law of nature whose confirmation has become routine in current particle accelerators.

Try to find the solution to this contradiction!

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

Is hydrogen atom in bifluoride connected by two covalent bonds?

In chemistry, a valence electron is an electron that is associated with an atom, and that can participate in the formation of a chemical bond. In a single covalent bond, both atoms in the bond contribute one valence electron in order to form a shared pair. For instance, hydrogen atoms have one valence electron, while oxygen atoms have two. That is why a water molecule contains one oxygen and two hydrogen atoms that are connected by singles covalent bonds:
Space filling model of a water molecule

By Dbc334 (first version); Jynto (second version) - File:Water-3D-vdW.png, Public Domain, https://commons.wikimedia.org/w/index.php?curid=1332739

So the Lewis structure of water is H-O-H.

But let us consider a more complex example: bifluoride. Bifluoride is an inorganic anion with the chemical formula [HF2].
Public Domain, https://commons.wikimedia.org/w/index.php?curid=1496287

It is not a strange anion. Some [HF2]- salts are common, examples include potassium bifluoride (KHF2) and ammonium bifluoride ([NH4][HF2]).

As shown in the figure above, the structure of the anion is symmetric, with the hydrogen situated in the mid-point of the F-F distance. In addition, the H-F contacts in this ion are very short, like in covalent bonds, and the corresponding H.F interactions are also strong enough to be classified as covalent bonds. So the corresponding Lewis structure of the anion should be [F-H-F]-. But, how is this possible? Hydrogen atoms have one valence electron. They cannot be connected by two covalent bond!

Are you able to give an explanation? Accept the challenge!

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

Is expansion of gases a violation of inertia law?

The first Newton's law (also called inertia law) says that, when viewed in an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a net force.

Let us suppose that we have a box with a wall splitting it in two halves. The left half contains a rest gas (no wind inside), while the right half is empty. Suddenly, we remove the wall. Because of the fact that gases expand to fill their containers, we know that the gas of the left half will move to the right side in order to fill the whole box:
But there is no net force acting on the gas, so why has it moved to the right half of the box? Do gases violate the inertia law?

Are you able to give an explanation? Accept the challenge!

Can we obtain extra surface by assembling the pieces in a different way?

Picture A and B show different arrangements made of similar shapes in slightly different configurations:
Missing square puzzle-AB
By Krauss (Own work) [CC BY-SA 4.0 (http://creativecommons.org/licenses/by-sa/4.0)], via Wikimedia Commons
So the question is: where does the extra one unit square come from?
Missing Square Animation
By Trekky0623 at English Wikipedia (Transferred from en.wikipedia to Commons.) [Public domain], via Wikimedia Commons
Are you able to solve this puzzle?

If nuclear fusion is the reverse of fission, why is energy released in both processes?

Nuclear power is the use of nuclear reactions that release nuclear energy to generate heat. There are basically two ways to release energy from nuclei:
  • nuclear fission, which is either a nuclear reaction or a radioactive decay process in which the nucleus of an atom splits into smaller parts (lighter nuclei).  This released energy is the one that is frequently used in steam turbines to produce electricity in nuclear power plants.
Public Domain, https://commons.wikimedia.org/w/index.php?curid=486924
  • nuclear fusion, which is a reaction in which two atomic nuclei fuse to form a heavier nucleus. Nuclear fusion reactors are not yet economically viable, but this technology is currently under research and it could become viable in a few decades.

Deuterium-tritium fusion.svg
By Wykis - Own work, based on w:File:D-t-fusion.png, Public Domain, https://commons.wikimedia.org/w/index.php?curid=2069575
If splitting a nucleus into two smaller nuclei releases energy, it seems that combining two smaller nuclei into one larger nucleus would require energy, not release it, because it is the inverse process. So, why can we obtain energy from both processes?

 Are you able to resolve this cognitive conflict?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

Why are cognitive dissonances useful for teaching?

Psychologists define cognitive dissonance as the mental stress or discomfort experienced by an individual who holds two or more contradictory beliefs, ideas, or values at the same time, performs an action that is contradictory to one or more beliefs, ideas, or values, or is confronted by new information that conflicts with existing beliefs, ideas, or values. It has been found that an individual who experiences a cognitive dissonance tends to become psychologically uncomfortable, and is motivated to try to reduce this inconsistency [Festinger, L. (1957). A Theory of Cognitive Dissonance. California: Stanford University Press].

That is why psychologists have incorporated cognitive dissonance into models of basic processes of learning, notably constructivist models [Ausubel, David P.,Novak, J.D.,Hanesian, H. (1978) Educational Psychology: A Cognitive View (2ª ed.). New York: Holt, Rinehart and Winston.] [Ausubel, David.P. (2000). The Acquisition and Retention of Knowledge. Dortrecht, Netherlands: Kluwer.].

In these models teachers gather information about students’ existing ideas and use this information to design activities that foster dissonances in their minds by increasing their awareness of conflicts between students’ prior beliefs and new information (e.g., by requiring students to defend prior beliefs and confronting them with unexpected experimental results or different ideas). Then, teacher guides students to find by themselves correct explanations that resolve the conflict.

Notice that with this methodology no external reward is needed. Cognitive dissonances are enough to increase students' enthusiasm for educational activities. Some researchers have concluded that students who are forced to attribute their work to this intrinsic motivation came to find the task genuinely enjoyable [Aronson, E. (1995). The Social Animal. New York: W.H. Freeman and Co.].

Once students have found by themselves the correct explanation that resolve the cognitive dissonance, a conceptual change has happened. Now, students do not see the world with the same eyes as before, and this change will remain forever, so they keep easily what they have learned even several years before. In fact, it has been shown that teaching methodologies based on cognitive dissonances significantly increase learning in science and reading [Guzzetti, B.J.; Snyder, T.E.; Glass, G.V.; Gamas, W.S. (1993). "Promoting conceptual change in science: A comparative meta-analysis of instructional interventions from reading education and science education". Reading Research Quarterly 28: 116–159.].

Therefore, we can say that we have found a magic wand that will make the students learn science! Nevertheless, things are not so easy. The really difficult part of this process is to find the suitable cognitive dissonances that make the students learn every specific chunk of scientific knowledge. And this is a too technical task to be left to psychologists! Scientists should do it. That is why I have created this web site, to make a database of questions/interventions that could help teachers to foster dissonances in student minds. Of course, since we are looking for cognitive conflicts with students beliefs, the way we can foster these conflicts depends strongly on the students. So,
  • If you are a teacher, you have to select the most suitable question/intervention in each case. You can also collaborate with this project by sharing the questions/interventions that you use to foster dissonances in your students.
  • If you are a student, enjoy trying to resolve the conflicts. You can post your trials in the comment box below each puzzling question. Some weeks after each publication we will include the explanations that eliminate the inconsistency.  
I hope you found this web site useful!

Sergio Montañez
Founder of Cognitive Dissonances