why does a broken neck kill someone?

You have 7 cervical vertebrae (C1-C7). Each one of them has a major nerve that branch off of it. (Plus an eighth nerve which is superior to C1).

Spinal nerves at C3, C4, and C5 are very important. (In the field we have a saying, "C3,4,5 Stayin' Alive"). The phrenic nerve originates here. The phrenic nerve is what controls your diaphragm, and therefore your ability to breathe. If this nerve is cut off, it can stop your breathing. There are also parts of your cardiac plexus (which controls the sympathetic tone of your heart....increased heart rate and force) that originate here.

Also, besides the risk of certain spinal nerves being cut off, you also have the problem associated with the spinal cord itself being damaged. One result of this can be Neurogenic Shock. This is when all the sympathetic signals that tell your smooth muscles in your blood vessels to remain "tight" for good circulation are cut off. This results in blood pooling and terrible circulation know as "relative hypovolemia". (Think about plumbing in your house. If the size of all the plumbing increased, but the amount of water remained the same, you wouldn't be able to get the water where you wanted it to go, right?) You can see this in a patient as a pale and cool skin area literally at the line below where the particular damage has occurred) Spinal cord damage can also be as "simple" as paralysis.

Depending on the severity all of the above may be temporary or permanent conditions. Also, you can even break your neck and have no consequences! (You would be very lucky, but still in deep trouble, as any further movement can result in the above scenarios.

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RE:

why does a broken neck kill someone?

Why does a broken neck kill someone, but other times it paralyzes the person, instead?

The nerves get damaged. That's why you get killed when you're hanged. Sometimes this doesn't happen and they'll have to wait untill they choke.

IT depends on how badly it effects the nerves. Don't forget the nerves and backbone connect to the brain, so a snapped nerve in the neck could easily shut off your brain