Automobile Airbags. 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 automobile accident with an initial speed of 60 km/h (37 mi/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?
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To determine the distance over which the airbag must stop you for you to survive the crash, we can use the concept of average acceleration.
First, let's convert the initial speed from km/h to m/s. By using the conversion factor 1 m/s = 3.6 km/h, we find that 60 km/h is equal to 16.67 m/s.
Next, we need to find the deceleration (negative acceleration) required to bring you to a stop. We can use the kinematic equation v^2 = u^2 + 2as, where v is the final velocity (0 m/s since you're coming to a stop), u is the initial velocity (16.67 m/s), a is the acceleration, and s is the distance. Solving for acceleration gives us:
0^2 = (16.67 m/s)^2 + 2a * s
Rearranging the equation, we have:
2a * s = - (16.67 m/s)^2
Now we know the magnitude of the maximum acceleration allowed is 250 m/s^2. Therefore, the maximum acceleration is -250 m/s^2 (negative since it's deceleration) as we want to stop.
Plugging in all the values into the equation, we get:
2 * (-250 m/s^2) * s = - (16.67 m/s)^2
Simplifying, we have:
-500 m/s^2 * s = -278.89 m^2/s^2
Dividing both sides of the equation by -500 m/s^2, we get:
s = -278.89 m^2/s^2 / -500 m/s^2
Now we can cancel out one of the units (m/s^2), and the remaining units (m^2) indicate distance squared. To get the actual distance, we need to take the square root of the value:
s = √(278.89 m^2 / 500)
Evaluating the square root, we find:
s ≈ 0.94 meters
So the airbag must stop you within approximately 0.94 meters (or 94 centimeters) for you to survive the crash.
V^2 = Vo^2 + 2a*d
Vo = 60,000m/3600s = 16.7 m/s
d = (V^2-Vo^2)/2a
d = (0-16.7^2)/-500 = 0.56 m.