Surviving a Fall

Girl falls 14 floors down chimney, saved by pile of soot. Her only injury was a broken hip, and it was guessed that she had fallen headfirst down the chimney and landed on her back.

No problem (with a bit of luck), but a good physics problem.

One "floor" is usually about 10 feet or about 3 m, so she fell about 140 feet or about 42 m, plus or minus.

Her maximum speed at impact would be sqrt(2 * 9.8 * 42) = 28.7 m/s, which is about 64 mph or about 100 kph.

Her actual speed will be much less, for two reasons. One would be any friction impact with the sides of the chimney, and the other would be air drag, which would be significantly increased by being inside a restricted area. It is more difficult for air to flow around an object when it has to squeeze into the gap between here and the walls of the chimney.

Her maximum deceleration at impact would be 42/0.6 (or 140/2 if working in feet) = 70 Gs (70 times the acceleration of gravity), which is a survivable impact under the right conditions.

Update:
Two other news stories(AP and New York Times put the height at about 180 feet (almost 55 m), which would increase the deceleration to about 90 Gs. That really is lucky.

The Times says the building has 13 stories (so they probably skip one floor and count 10, 11, 12, 14) plus a 25 foot chimney. That would be about 155 feet, not counting the roof height, but you also have to include the basement. However, that story also says "several feet" of soot, and that would have been after she landed and compressed the ash. Two feet is probably a lower limit on the distance to stop.

The Times said the chimney had an "opening of about five square feet", which would only be 27 inches square or 24" by 30". That must be wrong. She could easily touch the walls in that case, and would have trouble landing on her back. Their (innumerate?) writer probably meant "five feet square". That would leave room to rotate your body but increase drag due to restricted air flow around her body. (See my earlier comments above.)

It is kind of cute that we don't need to find the velocity to calculate the rate of deceleration. Both calculations use Vf² - Vi² = 2*a*X, so we have 2*g*Y = -2*a*X, so a = -g*(Y/X) with Y=distance fallen and X=distance to stop.

A good reference point for surviving collisions are the numbers used for crash testing of automobiles. NHTSA uses about 55 Gs at the chest as a fairly conservative limit on the acceleration your body can take in a crash. The LD50 number (the acceleration that will kill half of a population) is thought to be closer to 80 Gs with a torn aorta as the cause of death. [Your body stops but the heart keeps moving, tearing off the major blood vessels that are still attached to your body. IIRC, something like this is what killed Princess Di, who died of internal bleeding.] I say "thought to be" because accelerometers often measure values like 100 Gs in racing crashes where the driver walks away, although those drivers are in a form-fitting seat rather than landing in a pile of soot.

The acceleration of this girl on landing would be reduced because there were other forces acting on her as she fell, but she was extremely lucky that she landed in a way that applied the forces to parts of her body that are less vulnerable than her head. Breaking her hip absorbed energy that would have done greater damage if applied elsewhere.

Safety hint:

A 30 mph car crash without a seatbelt or airbag is like jumping off of the roof of a 2-story house and landing face first on the padded dash of an automobile that is lying in the yard.