Temperature is the amount of kinetic energy in a system. Are we feeling the force of the molecules hitting our skin when we detect something as being hot? Do we feel the kinetic energy in our skin decreasing when something feels cold? What do we actually feel when we detect temperature differences between our bodies and the objects with which we come into contact?
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You can’t actually feel outside temperature. If you hold your hand in water that is at 20°C / 68°F, it will feel quite cool, whereas air of the same temperature feels pretty neutral (or at least not nearly as cool). Similarly, a piece of metal that’s colder than your skin feels much colder than a piece of wood that is at the same temperature.
This is because, what you actually feel is the temperature (and changes in temperature) in your skin. That may sound the same, but it actually makes a big difference, because it means that for something to feel cold, it has to cool your skin. And the thing is, not all materials are equally good at transferring heat. Water, for instance, is quite good at this. Cold water next to your skin will readily absorb heat and cool your skin, while warm water will readily transfer heat to your skin. Metal is also very good at this. This why we make pans out of metal (among other materials), and boil things in water. But something like wood is very bad at this, which is why on a hot, sunny day you can safely sit down on a wooden bench. Air also does not transfer heat very well, which is very fortunate for us because it means we can tolerate quite a wide range of air temperatures around our bodies.
(Strictly speaking, there are actually two relevant properties: thermal conductivity and heat capacity. Thermal conductivity refers to how easily a material transfers its heat. Heat capacity refers to how much heat the material can hold. For instance, you need to add much more heat energy to a kg of water to heat it by 1°C than you need to heat a kg of air the same amount (about 4 times more). Water actually doesn’t conduct heat all that well, but it does have a very high heat capacity, so it still ends up delivering or absorbing a lot of heat when it is at a different temperature to your body.)
If you train yourself, you can get very good at estimating the temperature of a given material (say, water) based on how it feels, but that skill won’t transfer to judging the temperatures of other materials, because they transfer heat at different rates.
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