Re:Why your child's school bus has no seat belts
by Jach on Wednesday December 29, 2010 @03:17PM
(#2050008)
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.)
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.)