newtonian mechanics - Classical car collision

I have a very confusing discussion with a friend of mine.

2 cars ($car_a$ and $car_b$) of the same mass $m$ are on a collision course. Both cars travel at $50_\frac{km}{h}$ towards each other.

They collide. Ignoring any shreds and collateral damage, what is the speed of collision that the driver of $car_a$ felt?

What I mean is, if $car_a$ were to be driven into an infinite mass wall, what would the velocity be to replicate the damage caused by the initial collision?

Answer

I think you are asking how much damage would be done to the driver in the two cases you described. If that is your question, then the single car that is driven at a speed of $50\frac{km}{hr}$ into an infinite mass wall would experience the same damage as two identical cars being driven exactly head on at a speed relative to the ground of $50\frac{km}{hr}$ into each other.

You can easily understand this if you imagine an infinitely thin wall between the two colliding cars at exactly the plane where they collide. Assuming no shards or other pieces breaking off and going through this imaginary wall you can see that this is exactly the same as having an infinitely massive wall in place of either car since everything is exactly symmetric about this infinitely thin imaginary wall.

To be absolutely correct, this answer would actually require that the cars be left-right symmetric so that the centers of mass exactly line up perpendicular to the plane of the collision. If they were asymmetric it would be similar to symmetric cars hitting slightly off center - so there would be some torque around the point on the collision plane where the line determined by centers of mass intersect the plane. Clear?

EDIT: A situation which is equivalent to 2 cars hitting head on at 50kmph is the following: one car sitting stationary (with brakes off) while the other car hits it head on at 100kmph. This assumes a perfectly inelastic collision so that the two cars will then proceed (joined together) in the original direction of the 100kmph car but they will both be going at 50kmph. In both of these cases the change in speed of each car is 50kmph in a short time so the damages will be equivalent (either 50->0, 100->50 or 0->-50). However, if a car that is traveling at 100kmph hits an infinite mass wall the change in speed in a short time will be 100kmph so it is not equivalent to the two cases.