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

Venus, by NASA or Ricardo Nunes - http://www.astrosurf.com/nunes/explor/explor_m10.htm, Public Domain, https://commons.wikimedia.org/w/index.php?curid=338424

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.

By Original image by Niko Lang SVG version by User:Booyabazooka - Own work, CC BY-SA 2.5, https://commons.wikimedia.org/w/index.php?curid=5502912

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 (ν_e), 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 N_o 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 N_o, 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.