Analysis of an accident

imageA 800 kg car driving at 60 miles/h or 26.8 m/s loses traction in a curve and hits the wall of a house.  When it hits, it has slowed down to 40 miles/h or 17.9 m/s.  It breaks through the wall and comes to rest in the living room, 2 m from the wall.

The kinetic energy of the car is ½(800 kg)(17.9 m/s)2 = 128160 J.  The kinetic energy of the 60 kg driver inside the car is ½(60 kg)(17.9 m/s)2 = 9612 J.  The average force doing negative work on the driver over a distance of 2 m, if he is securely strapped into his car seat, is F = (9612 J)/(2 m) = 4806 N.  His average acceleration therefore is a = F/m = 80 m/s2 = 8.2 g with g = 9.8 m/s2.  This is probably a survivable accident.

The momentum of the driver is (60 kg)(17.9 m/s) = 1074 Ns.  To change his momentum to zero, this average force must act over a time interval ∆t = ∆p/F = 0.22 s.  The car comes to a stop in a very short time interval.

If the driver does not wear a seatbelt, he will initially keep on moving forward at 17.9 m/s.  In 0.1 s he will have covered a distance of approximately 1.8 m.  The distance the car has covered in 0.1 s is approximately x = v0t - 1/2at2 = 1.4 m.  If he sits initially 40 cm from the steering wheel, then his body will slam into the wheel and his head will slam into the windshield after approximately 0.1 s.  The car's speed after 0.1 s is approximately v = v0 - at = 9.9 m/s, so the driver slams into the steering wheel with a relative speed of 17.9 m/s - 9.9 m/s = 8 m/s = 18 miles/h.  If after an additional 0.02 s he travels with the speed of the car,  v = v0 - a(0.12 s) = 8.3 m/s, then his momentum has changed from p = (60 kg)(17.9 m/s) to p = (60 kg)(8.3 m/s) in 0.02 s.  This requires a force F = ∆p/∆t = 28800 N and an acceleration of 49 g.  Now the accident is probably no longer survivable.

What about airbags?

Airbags are designed to be leaky cushions.  When a car comes to a stop in a very short time interval in a crash, the driver or passenger will keep on going forward with approximately the same speed that the car had before the crash.  If the occupants of the front seats are not wearing seat belts, they will collide with the steering wheel, dashboard or windshield after a very short time interval.  Airbags are supposed to soften the impact by slowing the person down.  The person is supposed to make contact with the airbag after it is fully inflated.  As the person pushes against the airbag, the bag pushes back slowing the person down.  But the force with which the bag pushes back decreases rapidly, because the bag is leaking air and is rapidly deflating.  It is supposed to deflate in 1/3 of a second.

Because the person is supposed to hit the airbag when it is fully inflated, the airbag must inflate very rapidly.  The initial explosion in the airbag triggered by the crash must therefore produce a high pressure gas, which then expands rapidly.  When the sensor activates the airbag in a collision, a mixture of chemicals is ignited through an electrical impulse.  This causes a relatively slow kind of detonation which liberates a pre-calculated volume of nitrogen gas.  This gas fills the airbag in approximately 1/20 second.  If the pressure is not high enough, the airbag will not grow big enough in 1/20 of a second.

Nearly all airbag accidents happen when a person comes in contact with the airbag while it is inflating.  The airbag will then not provide a cushion, but hit the person with appreciable force.  Smaller people, who have their car seat pushed forward, are likely to come into contact with the airbag while it is still inflating, because then there is only a short distance between the person and the compartment where the bag is stored.  Rear facing child seats are also close to the airbag compartment.  When the bag explodes, it then hits the occupants with full force.  The airbags are designed to contact the chest area of an average size man.  If an exploding airbag hits a person in the chest, the likelihood of injury is not very high.  But people that have the seat pulled forward are often shorter that average, and the exploding bag is likely to hit them in the neck or head.  Now the likelihood of injury increases.

Reference: How airbags work

Problem:

A car moving at 10 m/s crashes into a tree and stops in 0.26 s.  Calculate the force the seat belt exerts on a passenger in the car to bring him to a halt.  The mass of the passenger is 70 kg.

Solution:
Let the car move in the x-direction.  For the passenger we have
∆p = pf - pi = 0 - 700 kgm/s = Favg∆t. 
Favg = -(700 kgm/s)/(0.26 s) = -2692.3 N. 
The minus sign tells us that the force point in the -x direction.