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).

Since the two cars have equal and opposite impact forces for the same amount of time (Δt) they must have equal and opposite changes in momentum. OR we could say that the total momentum BEFORE the collision must be equal to the total momentum AFTER the collision.

This is conservation of momentum. Notice that I haven’t said anything about the kind of collision. It doesn’t matter. Momentum will be conserved as long as the force on A is equal and opposite to the force on B. If one of the cars has an external force on it then momentum will NOT be conserved.

50–50 Head-on Collision

But if we want to look at the damage to a car during a collision, we need to consider energy. Imagine that we have two equal mass cars moving towards each other and they have a head on crash. After the crash, both cars are stationary. Here’s an animation.