Forces in a Collision

Is it worse being in a car crasing front to front with an identical car at speed v, when compared being in a car crashing with a solid wall at twice the speed?

Why

Hope you can use one minute to explain this mathematically.

The damage to a person involved in a collision arises from the forces to which the body is subject. The average force a body experiences is equal to the change in momentum of that body divided by the time over which that momentum change took place. A body's momentum is the product of its mass times its velocity. So for a person of a certain mass traveling at a certain velocity, the key to determining damage is the time it takes that person to come to rest during the collision.

Now we need to look at some of the real world details of an automobile collision. The time from the beginning of the collision until all is at rest depends on the structure of the car. Automobiles are not perfectly rigid objects so when the first part of the car to strike the obstruction stops, the rest continues on a bit. This accounts for the crumpled appearance of the vehicle after the collision. The effect of this crushing is to spread out the collision time for objects near the back of the car, reducing the damaging force they experience. If a passenger is unrestrained, he does not slow down until he comes in contact with a part of the car that has already slowed or perhaps stopped. This is not a good thing. For such a passenger whether the car hit an identical vehicle or a concrete wall probably does not matter.

For purposes of this discussion let us assume our passenger is fastened securely in his seat so that he slows down at the same rate as the middle part of the car. Now if his vehicle hits an immovable and uncrushable object like a concrete bridge abutment, he will come to rest in the time it takes for his part of the car to be stopped. Suppose now instead of a concrete wall we run our subject into another similar vehicle. To get equivalent damage, the time for the passenger to come to a stop would have to be the same. This requires that the point where the passenger comes to rest be fixed in space as was the case when he hit the wall. that means

that the other vehicle would have to have the same momentum as that of our subject's car.

The upshot of all this is that we should expect equivalent damage from a collision between two vehicles each with velocity of 100 kph as from a collision between a single vehicle at 100 kph with an immovable object. Your proposition that we hit a wall at twice the speed of a two vehicle collision should produce twice the force on the person in the car.