Robert Cohen reworded one of my paragraphs and asked if that is what I
meant. Yes, that's exactly what I meant. Thanks for saying it better.
Jim Green writes that we need to define the problem and stick with it.
I partly agree. But the problem is that collisions involving cars and
people are too complicated. It's not easy to transform our simple
air-track experiments and our billiard-ball equations into concrete
statements about injuries to people.
The original question was whether a head-on collision between two cars,
each going 35 mph, was equivalent to a car hitting a tree at 70 mph.
The answer is no. The collision between two cars each going 35 mph is
more equivalent to a car hitting a tree at 35 mph.
If we presume the question is attempting to assess the injuries to the
occupants of the car, that is, which situation would you rather be in,
then I would offer these answers: Pairing-one: a head-on collision
with another car, both going 35 mph, or a collision with a (large) tree
while going 70 mph. Answer: I'll pick the collision with the car... no
doubt in my mind here. Pairing-two: a head-on collision with another
car, each going 35 mph, or a collision with a tree while going 35 mph.
Answer: I don't know. They're roughly equivalent, and my chances of
survival in each case are based upon too many other variables.
The other car (or the tree) plus our brakes are what exerts the forces
on our car to bring it to rest. The collision of our body with the
steering wheel, dashboard, windshield, seat belt, air bag, etc. is what
brings our body to rest. This happens at slightly different timing
than the impact of the car. At the time our body impacts (let's say
the steering wheel) what is the steering wheel doing at that time? Has
it already come to rest? Is it still moving forward because our car is
still in the process of crumpling? Is it still moving forward because
we collided with a movable object and our car has pushed or is pushing
that object ahead of it, hence our car is still moving forward at
reduced velocity? Is it moving backwards because we collided with a
car or truck more massive (or going faster) than our vehicle?
We would prefer that the steering wheel is still moving forward when we
impact it. That would mean that our body's momentum does not have to
change as much when it is forced to match the velocity of the steering
wheel. If our body does not need as much momentum change when it hits
the steering wheel, it needs less impulse, and it receives less force.
If the front bumper of our car is going to come to rest at the point of
impact (hitting the tree or hitting a car with equal but opposite
momentum) we hope our car crumples a lot, because that might mean the
steering wheel is still moving forward when we hit it.
If the front bumper of our car comes to rest behind the original point
of impact (we hit a car going the opposite direction with more momentum
than we had) then the steering wheel might be moving in the opposite
direction (ground reference) than we, and this is clearly worse for us.
If the front bumper of our car comes to rest beyond the initial point
of impact, then we hit something movable, and there is greater chance
the steering wheel is still moving at appreciable velocity in our
direction when our body hits the steering wheel. That's clearly better
for us.
I don't see any way to turn this into any thing simple. We have to
ask, when our body comes into contact with the things that are going to
stop it, how fast are those objects moving, which way are they going,
what acceleration are they experiencing, how hard/soft are they, etc.?
These objects are all interior parts of the car (assuming we are not
ejected). These objects will not experience the same accelerations as
other parts of the car, because of crumpling. What part of the car
shall we use a reference point? The center of mass? The front bumper?
Although it might be best to use the center of mass, I like to use the
front bumper because it defines the point of impact, and the things
that support the things that our bodies hit are crumpling toward that
front bumper. But let's just say "the car."
What our inelastic collision equations mostly tell us is something
about where "the car" is going to end up. If we hit a stationary
immovable object, or an object with equal and opposite momentum, "the
car" pretty much stops. If we hit a stationary movable object, or an
object with less opposite momentum (or even with less same-direction
momentum) "the car" continues forward for a while. If we hit an object
with greater opposite momentum then "the car" gets pushed backward. It
seems to me that this paragraph is pretty straight forward inelastic
collision stuff.
The instantaneous changes in momentum, hence the instantaneous
impulses, our bodies experience depend on many variables such as what
internal car parts our body hits, whether we're wearing seat belts,
whether we have air bags, etc. BUT all of these variables still have
to act under the umbrella of the overall inelastic collision statements
made in the preceding paragraph. Thus, it is worse for us if our "car"
immediately stops than if it continues on a bit. And it is worse yet
if our car gets pushed backward.
Michael D. Edmiston, Ph.D. Phone/voice-mail: 419-358-3270
Professor of Chemistry & Physics FAX: 419-358-3323
Chairman, Science Department E-Mail edmiston@bluffton.edu
Bluffton College
280 West College Avenue
Bluffton, OH 45817