I was asked by an Applied Maths student of mine to explain an interesting Physics problem he came across online. Imagine two identical cars, one is at rest and the other is heading towards the stationary car at 10 m/s. When they meet there will be a small crash. Now the interesting part.

Imagine the cars are moving at 60 m/s and 50 m/s now. And you are above them in a helicopter moving at 50 m/s. In this reference frame the cars are still moving towards each other at 10 m/s. But when they meet and crash the accident is much worse. Why is this? Why, if these frames are equivalent, does the end result look so different? Let us look at each case.

Case I, you are watching the slow moving cars crash. Before they crash the Kinetic Energy is $\frac{1}{2}m10^2 = 50m$ joules of energy. After the crash the Kinetic Energy is 0 joules and the heat and sound energy is $Q_1$ joules and so $50m$ joules of Energy was converted into heat.

Case II, you are watching the fast moving cars from the helicopter. In this frame the total Kinetic Energy is still $\frac{1}{2}m10^2 = 50m$ joules of energy. What is energy after the crash? Well, it is not zero, after the crash both cars are moving away from the helicopter at a speed of 50 m/s. They have Kinetic Energy of $\frac{1}{2}m50^2 + \frac{1}{2}m50^2 = 2500m$ joules plus of course $Q_2$ joules of heat and sound (which we will assume is just a little more than $Q_1$). This is a gain of about $2450m$ joules, this gain must have came from somewhere. This came from the cars. The gained energy came from distorting the cars further. Distorted cars have more negative Potential Energy so to speak. If you are not happy with the idea with negative Potential Energy, then, you can consider each car having positive potential energy at the start which is loses at the end. This energy gives the car its shape and firmness (like a compressed spring) which it loses when it loses its potential energy.