Remember kids... always wear a helmet. (The almighty bikeradar helmet thread)

Appears to be a very common misunderstanding about the often quoted 12mph impact speed helmets are supposedly reliable for. The velocity with which your head might hit the ground has nothing to do with the speed you're doing at the time, be it 5 or 50mph.

Indeed. An object dropped from a height of 1.8m will hit the ground at (9.8*1.8)^0.5 m/s =~ 4.2m/s =~ 9.4mph.

In most accidents/falls the horizontal component of the velocity won't increase the magnitude of the primary impact of head (helmet or not) and ground at all; it does cause extra slidy tarmac/gravel pain though but that's a whole different kettle of badgers and much less likely to cause a KSI.

Pedestrians falling over will have their head fall from roughly the same height which is why you might see a greater reduction in (sheer numbers of) KSIs from compulsory pedestrian helmet use (risk homeostasis and other negative factors aside - the same arguments against compulsion apply to pedestrians as to cyclists).

Impacts with other objects (other people, vehicles, being hit by a wing mirror, etc) and my first paragraph doesn't apply. Hitting the side of a van at 30mph is going to result in a bigger impact than hitting it at 5mph obviously. Hitting the bonnet of a car and going ballistic is another case where the physics gets more complicated and the impact will depend, to some degree, on the speed you were traveling.