Explain why a high-speed collision between two cars would cause more damage than a low-speed collision between the same two cars

The force equation can easily prove this.  F=ma.  This states that the force on an object is equal to mass times acceleration.  If the mass stays the same and the velocity of the cars increases than that means there is a larger force.  This is because in both cases the cars are stopping in almost an instant and the times of the crashes are theoretically identical.  Acceleration is the change in velocity over time.  If the velocity is higher with the same amount of time than that means there is a higher acceleration.  If you plug a higher acceleration into the force equation then you wind up with a higher force and in turn a more damaging collision.