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As everybody knows that Black objects are black because almost all the light that falls on them is absorbed into the material. Little or no light is reflected back toward our eyes, therefore we see black.

It is also known that, since black objects absorbs all the light they ultimately become heated-up..!

If we see the snow in visible light, It appears to be White. But in the Infrared, it appears to be black. Does that mean that the Snow will feel warm in Infrared lights?

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Both Infra-red and Visible light are different. (It's BAD to call IR-radiation as IR-light either...) Visible light is everything that you see with your naked eyes. But, IR - NO, you can't see 'cause it's not light and that's why your eye won't perceive it. It's what most vipers, pythons and rattle snakes see..! This is the reason why snakes could detect warm-blooded animals, even in dark burrows. Snakes having two sensory pits are even thought to have some depth perception in the infrared..! (We'd be like an enormous fire-ball )

Thus, Infra-red rays are used for detecting thermal radiation (Thermography). IR-rays are emitted by each & every hot body. Even WE, humans emit IR-radiation of about 10 microns (10 $\mu m$) wavelength. So, I could see you even if you're in total darkness just using an IR camera...

There are three types of Infra-red. Far-IR, Mid-IR and Near-IR. All used for a different purpose...
OK, Now back to the topic..! Hence, Snow is too cold to emit IR-radiation. Maybe, that's why Snow looks dark in IR camera. But, I don't think it's black all-time. It maybe black when it's dark but not when there's light in an IR camera.

This paper would be useful for Optical properties of snow... It says that Snow has a strong absorption bands in the near-IR spectrum...

Please have a look at Electromagnetic Spectrum for your light confusion... :-)

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It means that snow is not reflecting infrared. Maybe absorbing maybe letting them pass through.

Also keep in mind that snow may be cold enough to look like black.

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