Why does hot water from the shower head feel colder at a distance than it does close up?

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Does it cool or is it due to dispersion?

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Cooled faster due to dispersion 🙂
If you have larger surface area then the heat is transfered more quickly to the colder air.
So if your shower head had that mist setting then you would lose a lot of heat quickly and water reaching you would be almost cold as the surrounding air.

Either and both. The dispersed water means you are touching less water at once meaning less heat transfers to you. The water also cools as long as it is exposed to air so the farther from the shower head you check the more time the water has had to cool. It also cools faster because it is spread out and in contact with more cold air per drop of hot water.