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Re: momentum



Peter Vajk's analysis is the correct approach. Yes, he has made some
assumptions, but these assumptions have to be made. If a car hits a
tree off center, or if two cars collide in a manner that is not exactly
head on, then we get spinning, secondary collisions with other cars,
collisions with road-side barriers, etc., and depending upon the
terrain (e.g. ditches) we can get roll-overs. Especially with air
bags, the spinning, roll-overs, and secondary collisions can cause more
human injuries than the initial sudden change in forward momentum.
Therefore the only way to compare the immediate injuries from
colliding with a tree to colliding with another vehicle is to assume
exact head-on collisions, with the car(s) coming to rest at some point
along the straight line of the original motion. If we cannot assume
that, then ordinary mechanics is not going to offer much analysis about
the severity of injuries, because all the secondary motion and damage
is too difficult to ascertain.

Many writers to the list have discussed energy considerations. In this
type of problem, energy is pretty much not important. This is a
conservation of momentum problem, which also means we have to study
impulse. The damage to people comes because their momentum must
change. This requires an impulse (integral Fdt). It is the size of
the force that does or does not fracture skulls, break ribs, etc. If
we can reduce the F and increase the dt to get the same impulse, i.e.
the same momentum change, we save lives... and that's what air bags do.
This is important, so let me repeat it. We need to reduce the forces
exerted on the occupants when collisions occur. It is the magnitude of
the forces that cause the injuries. To study the forces it is easiest
to study momentum, not energy. Energy conservation is occurring, but
it's not the way to study the forces. To reduce the forces on the
occupants, we either have to reduce the momentum change (e.g. drive
more slowly), or we have to spread the impulse (momentum change) over a
longer time.

In a collision with a tree (or the proverbial brick wall) the car comes
to rest at the tree/wall. In a collision with a parked car both cars
move in the direction of the original motion of the moving car. Hence
when hitting a parked car the resting place is beyond the initial point
of contact. So even though the original car eventually comes to rest,
and the total impulse is the same, the collision is spread over a
longer time, hence the forces on the car and its occupants are less.
So it is better to hit a parked car than to hit an immovable wall or
tree.

In order to collide with a car and have the original car come to rest
near the point of impact (like it does with the wall or tree) the
second car must be moving toward the first car with equal momentum. If
the second car has less magnitude of momentum than the first, the
resting point will be beyond the point of impact ("beyond" from the
viewpoint of the first car). If the second car has more momentum than
the first car, the final resting point will be ahead of the impact
point (the first car is pushed backwards by the collision). Thus, only
when the second car has the same (but opposite) momentum will the
two-car collision be the same as the tree/wall collision. Hence, if we
have equal-mass cars each going 35 mph in opposite directions, both
cars come to rest real close to the initial point of impact. Therefore
the dt is essentially the same as the tree/wall collision. Hence the
head-on collision with the other car, both going 35 mph is indeed
equivalent to hitting the tree/brick-wall at 35 mph (not 70 mph).

If the two cars are both going 35 mph, but do not have the same mass,
the lighter car gets pushed backwards and the heavier car continues
forward from the point of impact. In the end, both cars have come to
rest. However, even though the lighter car eventually went from 35 mph
to zero, it actually had a negative velocity for a while, hence its
momentum change at impact was higher and the lighter car occupants
likely have more severe injuries.

Notice that energy has not been invoked in my discussion except for my
original statement that it is not important. Total energy is, of
course, conserved. Kinetic energy is clearly not conserved, the car(s)
always end up at rest. So the car(s) always lose all kinetic energy
regardless of what they collided with. The car(s) also eventually lose
all momentum. But momentum is a vector, as are the forces on the car
and occupants, so it makes a difference during the collision whether
the momentum goes to zero, or perhaps reverses, and how suddenly this
happens. Kinetic energy is not a vector, and consideration of energy
doesn't add any new information to the momentum/impulse/force
discussion in terms of injuries, etc. That is, we just don't get much
useful information from considering energy.

For those of you worried about the total kinetic energy being more when
two cars collide than when one car collides with a tree/wall... this is
true... but it's nothing to worry about. While it is true that the
two-car collision has double the energy to dissipate, we now have two
cars to crumple. Assuming the cars are equal, half the crumpling takes
place in one car, half in the other. The crumpling (i.e. KE to heat)
that takes place in either car is the same that would have occurred if
either car had collided with the tree/wall. Of course we are assuming
that the tree or wall is not damaged. As long as we consider a
substantial tree or wall, that is typically the case. If the tree is a
foot in diameter or larger, about the only thing a 35 mph crash causes
is some bark removal.

Incidentally, it is one of my pet peeves that many teachers who do the
"egg drop experiment" discuss this from a viewpoint of energy. That is
not the correct approach... it is clearly an impulse problem. We need
to accomplish a change in momentum without exceeding the force that the
eggshell can withstand. If the students understand momentum changes
and impulse, it is clear that all they need is a device to make the
collision with the ground last for a longer period of time. Or better
stated, the egg has to be brought to rest in sufficiently long time
that the force does not exceed the breaking force of the shell. (And
of course we assume the force is spread out over the shell rather than
occurring at one point.) Energy considerations don't add any insight
into this experiment at all. Although you can use the language that we
need the energy change to occur over a long time, that wording doesn't
lead to meaningful equations as quickly as realizing that delta-p is
equal to integral(Fdt).

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