How far do you have to be from a nuclear bomb to survive?
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Far too many variables to give you a fixed answer.
How big a nuclear explosion are we talking about?
A 22 kiloton “Nagasaki” sized atomic blast is not the same thing as a W-88 Thermonuclear warhead.
Also… safe from what exactly? the physical blast wave, the ionizing radiation pulse, the thermal pulse, fallout? Different effects are bigger than others given the size of the bomb Yield. The different effects do not scale the same so by way of example, is smaller detonations the thermal pulse is not as far reaching as the blast wave itself, but as you go bigger ad bigger, the blast wave tails off more rapidly with distance and does not scale up the same way as the thermal pulse. With the bigger nukes, you can be well outside the blast radius and still receive lethal thermal exposure and die of massive 3rd degree burns to your body if exposed to line of sight to the fireball.
For the blast itself it depends on what sort of shelter you are in. The blast gets progressively weaker the further you are and what may not be good shelter close in, would be adequate further out. far enough out and you can survive the blast wave in the open (provided you are not struck by debris thrown outwards from further in).
Don’t be downwind or in the immediate vicinity.
For a 1 megaton ground detonation, outside of about 10–12 miles you should be reasonably safe so long as you have some sort of decent shelter, (much the same as sheltering from a tornado). At 20 miles, the blast would barely break windows, don’t stand near them or you may be injured by flying glass.
Being outside however, you would still be in danger from the thermal pulse (heat wave) radiating from the fireball, but so long as you were out of direct line of sight of the fireball itself, you would be fine.
I would say that as long as you had some sort of physical barrier between you and direct line-of-sight to the fireball, you would be safe from just about every typical nuclear weapon out there as long as you were at least 20–25 miles from ground zero (and not downwind from fallout)
The above is a 1 megaton example. Most nukes are not that big.
The two most common nukes in the US inventory are 475 and 500 kilotons
500 kiloton is 1/2 a megaton.
It isn’t easy. Some believe that if all the nuclear warheads were launched the earth might be uninhabitable anyway. This would be true if the new version of the Russian nuclear bombs are implemented, which scatter radioactive materials which would circulate everywhere. But there is a reasonable possibility that a nuclear winter would not happen and if you survived the initial war and you lived in a nation that was not destroyed you might survive.
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As is true with most modern weapons systems, the answer is, it depends on a number of complex factors.
Just off the top of my head:
The kind of nuclear blast makes a huge difference. A dirty bomb will just scatter around radioactive dust in the immediate area. A modern hydrogen bomb can easily destroy an entire city with the initial blast alone; that’s to say nothing of the long term radiation danger. Problem at your local nuclear power plant? The immediate danger zone is almost certainly less than 10-20 miles from the plant.
If we’re talking about a “launched” nuclear warhead then the detonation altitude also becomes relevant. In general, the lower the detonation altitude the more “focused” the blast area will be. Nukes launched against reinforced military targets detonate at lower altitudes to do more damage in a smaller area. This allows the blast to be strong enough to destroy appropriately strong/shielded military buildings. Nukes launched against general civilian targets detonate higher because civilian buildings are frequently easier to knock down so spreading the force across a larger area results in more buildings being destroyed.
When in comes to the dangers of radioactive fallout time, location, and weather matter.
In general, it takes time for radiation to fall back to earth after a nuclear warhead is detonated; usually something between 12–24 hours.
In that case, assuming you survive the initial blast it’s key to head upwind of the larger area weather patterns and generally in the opposite direction of the jet stream (that usually means head West in the Northern Hemisphere, and head East in the Southern Hemisphere) before radioactive fallout starts coming down.
A major nuclear explosion from a military device will almost certainly result in retaliation, and then (potentially) further attacks and counter attacks by additional parties. At some point, this is expected to trigger a nuclear winter; killing plants, animals, and causing global temperatures to plummet to a new ice age. Between the multiple blasts and the immeasurable devastation of a nuclear winter event most people’s chances of survival are essentially zero.
Generally, with proper security clearances, you can get within inches of a nuclear weapon and be perfectly safe.
If the thing actually detonates, however, there is no reliable way to determine “safe".
Close enough, you won't have time to worry.
Far enough to survive the blast you will worry about if medical care survived and will reach you in time.