What is the critical deceleration at which a seat belt locks?

[jannieDelta]

Homework Statement

At some critical deceleration, a seat belt will lock into place, preventing a driver/passenger from pulling it out any further. What is that critical deceleration and why?

Homework Equations

I have not quite figured this out yet, because it would depend on how you go about solving it, wouldn't it? See below.

The Attempt at a Solution

We looked up seat belt locking mechanisms at howstuffworks.com -- there's two types of simple systems, one of which depends on the speed at which you're pulling out the seat belt itself, and one of which depends on the deceleration of the car.

At first we were thinking that the deceleration would have to be just low enough to ensure you don't get seriously injured when you hit the dashboard/seat back/airbag. However, this would involve knowing how much of a deceleration will leave you with an injury when you hit something, which seems like it would be rather complicated to figure out and would depend on way too many variables -- the angle of impact, the surface you're hitting against, what hits it first, etc.

We also considered the possibility that the function of the seat belt is to lock during a collision -- if you take a bunch of fairly plausible collisions, with different values for a number of variables (mass of either car, speed of collision, type of collision), and applied conservation of momentum and such, we could probably figure out the deceleration involved, and once we have a bunch of numbers... The problem with this is that it doesn't seem at all rigorous, and it still seems like there are too many variables to take into account.

Is there something simple that we're missing here? This is a question for a calculus-based physics class, and we need someone to point us in the right theoretical direction (if we understood how to do it, I don't think we'd have much trouble with the numbers) because at the moment we're rather lost.

 

[dotman]

Hello,

Not really sure what to tell you here-- you're right that the purpose of a seatbelt is to lock into place, and most contain pretensioners to draw up the slack. In conjunction with the airbag, the intent is to keep your body from hitting the steering column, windshield, other passengers, etc, which causes severe injury or death. They trigger when a sudden deceleration is sensed, although I don't know what that value is; presumably it's well below a value that would cause you any injury at all, and certainly below the air bag deployment value (I think we've all had our seatbelt lock up when we slam on the brakes, say, when a light goes red and you don't notice... but the airbag doesn't deploy).

I don't know if it will help, but you may want to check out this site for some (albeit simple) numbers:

http://hyperphysics.phy-astr.gsu.edu/Hbase/carcr2.html#cc1