Is a 50 mph-50 mph Collision the Same as a 100–0 mph Collision?

A long time ago there was a MythBusters episode that looked at head to head crashing cars. Suppose you have two equal mass cars both traveling with a speed of 50 mph and they crash (don’t worry, no one is in the cars). Would this cause the same amount of damage as a car going 100 mph and crashing into a wall? It’s time for some physics.

Basic 1D Collision

Suppose I have two objects (car A and car B) moving towards each other along the x-axis. When they collide, there is a force acting on both cars.

Since forces are always an interaction between two objects, the magnitude of the force that B pushes on A is EXACTLY the same (but in the opposite direction) as the force that A pushes on B.

Let’s assume that this impact force is the only force acting on both cars so that the net force on A is just F_B-A. What does a net force do to an object? It changes the momentum. I guess I should add that momentum (p) is the product of an object’s mass (m) and velocity (v).