How much more braking distance is needed at 60 mph than at 20 mph if you increase your speed from 20 to 60 mph (initial speed multiplied by three)?

A 3 x 3 or 9 times.

explanation

The faster you drive, the greater the impact or striking power of your vehicle. When you double your speed from 20 to 40 mph the impact is 4 times greater. The braking distance is also 4 times longer. Triple the speed from 20 to 60 mph and the impact and braking distance is 9 times greater. [Commercial Driver Handbook]