This one's not too hard. If you make the following assumptions about the two-car collision, it is easy:
1) The cars are identical (most importantly, they have the same mass)
2) In the collision, the cars stop at the place where they collide.
In a real collision, the cars may bounce off of each other at an angle,
spin around and come to a stop somewhere off to the side, or they may
even slide one atop of the other, and roll and turn. We'll assume they
just smack into each other and stay that way, possibly bouncing
backwards (an exact head-on collision).
Even if the cars are identical, if the engines are not in the
center but more to one side, corresponding parts of the cars will not
hit each other and the result could be a spinning, twisting wreck after
the head-on collision. We'll assume this doesn't happen.
Assuming these things, then the force as a function of time for a
driver in one of the cars will be the same for the head-on collision as
it is for the stationary wall. The reason is that because the collision
is symmetrical, and the cars do not penetrate into each other, their
pieces as they deform stay on one side of an imaginary wall separating
them. This is even true in the morbid case (but altogether too common
when the drivers forget to wear their seatbelts) that the driver is
thrown through the windshield (as he probably will be at 100 mph!) and
hit either the other driver coming the other way (I said things weren't
off-center!) or the solid wall.
In fact, the car crash tests that are done to simulate a head-on
impact with an identical car are simulated in exactly this way, with a
fixed wall, in order to halve the cost of the test.
In a real collsion, cars bounce, slide over each other, skid and
spin, and parts of one car do penetrate into the space normally
occupied by the other car, and so the collision with the other car will
probably be less hard on the driver than the collision with the solid
wall. Of course it will be worse for a driver if the other car is
heavier than his own.
(republished on 07/11/06)