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The human body can survive an acceleration trauma incident (sudden stop) if the magnitude of the acceleration is less than 250 m/s². If you are in an auto- mobile accident with an initial speed of 105 km/h and you are stopped by an airbag that inflates from the dashboard, over what distance must the airbag stop you for you to survive the crash?

So, I modeled the problem with this diagram:

enter image description here

It says that the initial position is $0m$ and initial velocity is $29.2\frac{m}{s}$. After $t'$ time has passed, the car crashes; it has traveled a distance of $X$, and since it crashed, velocity is 0.

Acceleration is fixed at its maximum value, which is $250 \frac{m}{s²}$

Now, I use this formula to obtain $X$: $$ v_x^2=v_{0x}^2+2a_x\left(x-x_0\right) $$

replacing, I get

$$0,00\,\frac{m}{s} = 852.6\,\frac{m²}{s²}+500\,\frac{m}{s²}X$$ so I solve for X:

$$ X = \frac{-852.6\,\frac{m²}{s²}}{500\,\frac{m}{s²}} = -1.70\,m$$

But it's kind of weird that I get a negative distance, so I check the solutions manual; what I get is that $1.70$ is correct, but direction is wrong (the negative sign):

enter image description here

The reason is that they choose a negative acceleration, but I don't get why. Could you explain to me why it has to be chosen negative?

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closed as off-topic by ZeroTheHero, Ben Crowell, Aaron Stevens, John Rennie, Jon Custer Jan 18 at 14:21

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  • $\begingroup$ Don't get all "hung up" on the negative sign. This is physics, not math. $\endgroup$ – David White Jan 18 at 1:31
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    $\begingroup$ I've added the homework-and-exercises tag. In the future, please use this tag on this type of question. $\endgroup$ – Ben Crowell Jan 18 at 4:12
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Hope you guessed why you get a negative distance by now. If you have not remember the human body survives if the acceleration is less than 250 m per second squared. In the example the trauma is caused by a deceleration. So if you had used negative value for acceleration you would have got the positive distance as expected.

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In order to stop, the velocity has to decrease from the ~30 m/s to 0. In order for velocity to decrease, the acceleration has oppose the motion: hence, the acceleration must be negative.

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