August 2008 Archives
> James -
>
> Hi there. Matthew Belinkie, YPMB '02 here. I have a physics problem,
> and I seem to recall that is (or was) your area of expertise.
>
> I want to calculate the distance a person would fall over a certain
> number of seconds. I know that's a pretty straightforward equation...
> if you assume you're in a vacuum. The tricky part is, I DO want to
> take into account wind resistance, terminal velocity, things like
> that. I want to know in the real world, if you fall out of a plane,
> how far will you travel in 25 seconds.
>
> More info: this is about a scene in Superman 2. Superman saves a
> little boy who falls off Niagara Falls. But this kid falls for
> literally 25 seconds before he's caught. The Falls are 167 feet high.
> I want to crunch the numbers and figure out how far a little boy would
> ACTUALLY fall in 25 seconds. (Let's assume the kid weighs 70 pounds.)
>
> Well James? What do you think?
>
> - Matt
>
Hey, Matt!
Here's a back-of-the-napkin answer.
Precise accounting for resistance in a fluid is almost impossible. The equations of motion in a fluid involve a factor called "B", which increases with velocity and is dependent on the viscosity of the fluid, the shape of the falling body, how the fluid passes over the falling body, qualities of the skin of the body where semi-laminar flow may occur, and the weight of the body. In practice, very fast computers crunch on the differential equations that result when you account for fluid resistance.
Considering a vacuum, a falling object will take 3.2 seconds to fall 167 feet. The body will be falling only 70.4 miles per hour, which is fast, true, but not quite terminal velocity. A falling person, depending on how they are falling (arms out, balled up, in between...) will reach a terminal velocity of between 200 and 125 mph. Assuming that the kid is terrified and crapping his pants, he probably won't think enough to spread out his arms and legs to catch the wind and extend his total travel time (and hence his life!) to around 4 seconds from the 3.2 he's have in a vacuum. Silly kids, never thinking!
In 25 seconds he can fall 10,000 ft in a vacuum, so perhaps 8,700 feet in an atmosphere (feel the back of the napkin, Matt...). I hope he took a deep breath before falling off that airplane into the oxygen starved air at 10,000 ft. Oh, at that height, the air is thin enough that helicopters have trouble operating. Recall that people don't get rescued from high mountains with helicopters?
Well, I hope you won your beer by getting the egghead to "prove" that movies lie on occasion. Besides, it's Superman! Disbelief? Check it at the door.
James
