Jach's personal blog

(Largely containing a mind-dump to myselves: past, present, and future)
Current favorite quote: "Supposedly smart people are weirdly ignorant of Bayes' Rule." William B Vogt, 2010

Have some physics! Impulse

In the interest of posting something I'm going to mostly copy a comment here that I made on a forum. :)

To set up some context: why do buses have big, foam-padded seats instead of seatbelts? The answer is that foam-padded seats provide the same function as a seatbelt, but without the risk of strangling a kid or breaking one of their bones. Buses rarely tip over (though there are some (and I may be going to hell for suggesting) humorous videos on youtube of such incidents), and it still seems better to let the kids fall to the roof than get hung by a seatbelt (possibly from their necks!!).

Anyway, what is it about seat belts and foam seats that make them desirable?

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

The time-averaged force [math]F' = \frac{1}{\Delta t} * I = \frac{1}{\Delta t} * \Delta p[/math] 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 [math]\Delta t[/math] needs to be longer (the collision takes a longer time, due to e.g. the foam seat, seatbelts, 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.) I wish I took pictures.

Impulse is really a simple idea, even if it requires basic calculus to derive, but actually seeing it in the context of forces rather than just mass and velocity (as a change in momentum) gives a lot of intuition into how we can improve the safety of collisions. We have the capability to make our vehicles out of cheap material that wouldn't even dent during a high-speed collision, unfortunately the people inside would experience such an Impulse that they wouldn't survive.

Posted on 2011-01-24 by Jach

Tags: math, physics


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