I witnessed a multi-vehicle crash last night, and was just reviewing my dash cam video. (You can see it [mp4 video, you'll have to download to view] at https://app.box.com/s/9f21n705od726fxjj1zbi8djiyvu92yo if you'd like).
The "middle" vehicle, a small pickup, was spun after being pushed by a vehicle passing him on the right. He hit a third vehicle 90 degrees through that spin. It took 25 frames (5/6 of a second) to spin another 270 degrees after the second impact. (Side note: the video above only shows about 10 frames a second, I'm counting off the original video at 30 fps)
For the sake of coming up with a best-guess value, I'm going to offer these assumptions:
- his speed at initial impact was 40 miles an hour
- at the point of second impact, he was moving at 5 miles per hour at a 90-degree axis to his original travel. The original spin had stopped and he was moving in a straight line.
- distance from the point of impact to the center of his mass was 10 feet
- (if it makes any difference, the truck he hit was moving 40 miles an hour at 90 degrees to the small truck's motion)
I have two questions:
- what was the linear speed of the driver, in mph, during the 5/6 of a second he spun 270 degrees?
- what force (pounds, units of gravity, whatever) did he undergo at impact (I'm thinking this is an instantaneous acceleration question)?
I understand the basics of the first calculation involve the radius (10 feet) and the radians he moved around the center point. But that's the limit of my understanding of this particular geometry problem. I have no clue how to calculate that instantaneous force.
The small truck driver was pretty dazed and was taken to the hospital. What I'm basically wondering is how badly his brain was shaken in his skull (coup-contracoup injury). I realize that's beyond the scope of this group, but with the two numbers calculated above, I can do some research on the head injury piece. It's all for interest sake, anyway...I don't know the guy.
This question really needs to be posted to [Physics.SE]
There are two components of acceleration (and forces) at play. One is the deceleration of the center of mass and the second is the spin and the centrifugal forces.
My estimation is that the front of the red truck came to a stop really fast (the truck center of mass remains near the impact location along the road). You can see the bed cover flying out at about the speed the truck had before impact. I also estimate about 40mph before impact and 0mph after impact. I think happened over 0.2 seconds, so the decelleration is
$$ a = \frac{40 {\rm mph}}{0.2 {\rm sec}} = 89.6 {\rm m/s^2} = 9.1 {\rm gee}$$
Now as far as the centrifugal acceleration you noted 270 degress in 5/6 of a second, which means
$$ \omega = \frac{270°}{0.833 {\rm sec}} = 5.65 {\rm rad/s} $$
We have to estimate the distance of the person from the center of rotation. To me it looks like the center of rotation is in the middle of the dash of the car. The driver got flung to the window which is like 2ft away ($r=0.61{\rm m}$). The acceleration due to the rotation is
$$ a = r \omega^2 = 19.5 {\rm m/s^2} = 2.0 {\rm gee}$$
So there you have it, 9 gee of acceleration due to stopping and 2 gee due to rotation.