Speeding increases energy in a crash proportional to the square

A car moving at 70 miles per hour has to stop suddenly. Another car going 100 miles per hour also has to stop suddenly. Your intuition might say that the former requires 30% less energy to stop, but the energy required is actually proportional to the square of the velocity. Ben Sparks for Numberphile explains:

Okay. Now what are the energy gains and losses for the guy trying to speed by weaving in and out of slow traffic?