While we're here talking about physics: another big reason why buses are safe are because of their high mass. I was in one accident while on a school bus - another car hit the bus head-on while the bus was stopped. The kinetic energy of the 1 ton car travelling at say 30 mph - 1/2 * m * v^2 - was absorbed by the bus. I won't step through the math, but because of the high mass of the bus, the resulting velocity of the bus+car was quite low (it also helped that the change in velocity was mitigated by brakes/tires/friction, as the driver was ready for the impact).

In any case, as a passenger on the bus, the result of the impact was quite minimal - there were no injuries on the bus, and the bus actually drove off afterwards (with no passengers and a bit of sheet metal scraping the pavement).

Granted, it's a different story if the bus hits a tree or is hit by a tractor trailer doing 70 mph.

Another reason why buses are safe? The bus is operated by a trained professional, most of which have years of experience driving the bus for hours every day.

I have often wondered about the difference in accident rates between professional and other drivers, anyone know if there are any studies done on this?

My own experienceL My school bus when I was young hit into the side of a van and sent it physically flying and rolling. We barely felt it in the back.

To compliment your explanation, here's some math.

Newton's law: F = m* a = m* dv/dt. If we assume the child and bus seat's masses are constant (no exploding kids, please!), this leads to F = d(m* v)/dt = dp/dt where p is momentum. When the child hits the foam seat, the force will not remain constant, that is it varies with time, leading from the above to dp = F* dt. We can find the change in momentum by integrating Force with respect to Time, leading to (in LaTeX notation) p_f - p_i = \int_{t_i}^{t_f}(F* dt) = I where I stands for Impulse.

The time-averaged force F' = 1/delta(t) * I = 1/delta(t) * delta(p), and this is where I think it becomes very clear that we like to have longer collision times for less damage. If we're interested in having a smaller average force for when the child hits the seat, then either delta(t) needs to be longer (the collision takes a longer time, due to e.g. the foam seat, the bumpers on cars, collapsible sections of the vehicle, etc.), or the change in momentum of the child needs to be smaller (e.g. if the collision is inelastic and the seat brings the child's velocity to 0, that is better than the seat bouncing the child back). It's easy to add another layer of foam.

Possibly the funnest lab in my high school physics class: shooting an egg at a brick wall and having it survive unharmed by encasing it in some design. (Mine was the Hindenburg of Bubblewrap.)

The term you are looking for is reduced/exaggerated "impulse" which is defined as change in momentum aka delta(p) = F * delta(t).