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This is a problem that has many answers on google but I don't understand what is the underlying logic of the steps taken.

Here's the problem

  • The human body can survive an acceleration trauma incident if the magnitude of the acceleration is less than $250~\mathrm{ m/s^2}\;.$ If you are in an auto-mobile accident with an initial speed of $100~\mathrm{ km/h}$ and are stopped by an airbag that inflates from the dashboard,over what distance must the airbag stop you for you to survive the crash ?

Now,my problem is that I find it really confusing to understand what distance they're meaning. Let me explain.

The first time I've read the problem I was thinking that it was asking me the distance between me,the driver, and the dashboard ,so the point of the problem was to calculate the distance where the airbag should impact with me at an acceleration less than $250 ~\mathrm{m/s^2}\;.$

However I've seen on youtube a video about this same problem which was solved by finding the distance between the starting point of the car with initial speed $V_i$, and the point of the incident where $V_f=0\;.$

This really confuses me because I am not understanding the logic of this.

  • The airbag will inflate only at the point of the crash so why do I calculate the distance the car has travelled before the incident?

  • Does the airbag know that I will crash so it stops me before the incident?

My idea to solve the problem

My idea to solve the problem was that I needed to define a coordinate system at the time the incident occurs. So I would need to know the time it takes the car to go from its initial speed,$V_i$,to $0$,so that I know what is my resulting velocity at the time of the crash and finally calculate the point in my coordinate system where the airbag should impact with me at an acceleration less than $250~\mathrm{ m/s^2}\;.$But this is wrong apparently.

  • Can you guys make it clear what's going on here? What conceptual mistake am I making here ?
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    $\begingroup$ Ignore the car. You have a human traveling at $100km/h$; how far will the human travel in the process of slowing to a halt at $250m/s^2$? $\endgroup$ Commented Dec 26, 2015 at 14:09
  • $\begingroup$ Why does that answer my question?After the human has traveled that distance,the airbag will inflate.I am not seeing the connection. $\endgroup$
    – Mr. Y
    Commented Dec 26, 2015 at 14:19
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    $\begingroup$ You are slowing the person to a halt by the surface of the airbag. It doesn't matter what is on the other side of the airbag, or how it is moving. The human's (and the airbag surface's) deceleration is all that matters. $\endgroup$ Commented Dec 26, 2015 at 14:38
  • $\begingroup$ That makes it clear.So If the human somehow escapes the airbag at some instant $t$ during the slow-down-interval his speed will be however too much high that his collision with some object will make him die ,right?(However I still can't see how the human can be stopped in $1.7$ meters(that's the solution of the problem) in the space between him and the dashboard). $\endgroup$
    – Mr. Y
    Commented Dec 26, 2015 at 14:51
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    $\begingroup$ By way of improvement, I think if you were to delete the "My idea to solve the problem" heading and everything afterwards, you would be left with quite a good question, and I would favor reopening it. (Well, the title would also need editing, but that wouldn't be a reason to keep it on hold.) In its current version... I'd have to leave it to one of the people who voted to close to explain why they did so. $\endgroup$
    – David Z
    Commented Dec 27, 2015 at 9:56

1 Answer 1

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The question is ill-defined; it does not specify the frame of reference for the distance to be calculated. Is it the distance in the car frame (from seat back to steering wheel), or the distance on the ground the driver travels?

Assume that the car is extremely strong; it hits a granite mountain and comes to an instantaneous stop. Then the airbag must stop the driver, travelling at the original car speed, before he reaches the now stationary dashboard.

OTOH, if the driver presses gently on the brakes, the car may take several kilometres to slow, and the airbags would not even be needed.

Note that the airbags don't stop the driver; they keep the driver firmly but safely attached to the car, while the car slows down. Engineers have worked hard to change the "extremely strong" car above into a moving crumple zone, where each part of the car (bumper, grill, radiator, engine compartment, fire wall) collapses in turn while slowing the passenger compartment at a survivable rate. ' So the question could be rephrased: How far will, say, the driver's seat headrest move while the crashing car slows from its initial speed to zero at $-250\text{ m/s}^2$. The airbag will keep the driver in step with the headrest. The answer will be the amount of crumpling in the car front end...

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  • $\begingroup$ Thank you for this answer.This really clears out any ambiguity from the question. $\endgroup$
    – Mr. Y
    Commented Dec 26, 2015 at 16:06
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    $\begingroup$ I disagree that the problem is ill defined. The deceleration of the passenger has to be taken in the frame of reference of the earth (where "come to a halt" makes sense, in the collision sense of the word), and it's the distance traveled in that frame of reference that determines the instantaneous acceleration experienced. Also an air bag is mostly intended to increase the area available for supporting the passenger during deceleration - seat belts alone can cause significant injury, see for example faa.gov/data_research/research/med_humanfacs/oamtechreports/… $\endgroup$
    – Floris
    Commented Jan 2, 2016 at 19:39

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