In his memoir, Feynman described figuring out the equation that described the curve of water flowing from a tap with the valve slightly open. How would you check something like that? That curve looks like a lot of functions.
Thanks,
Rob
Obviously there are a lot of variables involved but I did find this:
If the water is flowing straight down as it comes out of the faucet it will continue straight down. Suppose when it exits the faucet it has a horizontal component of h and a vertical component directed downward of v to its velocity. It will in the absence of air resistance keep the same horizontal component so its displacement from (0,0) at the faucet exit at time 0 will be x = ht horizontally and y = vt + 0.5gt[sup]2[/sup].
So the equation of the curve is y = (v/h)x + 0.5gh[sup]-2[/sup]x[sup]2[/sup]. This is a parabola.
From your description I suspect that Feynman was not yet a practicing physicist when he figured this out since it would be hardly noteworthy if her were so I’d guess this is what Feynman determined using high-school physics. If he took air resistance into account, then he probably had more physics at this time. I seem to recall the restraining force of friction due to air resistance is proportional to the velocity squared, but am not sure. In any case, you’d use that force and F = ma to alter the acceleration. Note this does not require knowing the mass m because both directional component accelerations will be proportional to m so it will not affect the curve’s shape.
I’d imagine that the working assumption was that the water was maintaining a continuous stream rather than breaking up. Since the amount of water per time is the same at any cross-section of the stream, and the water further down is moving faster due to the acceleration of gravity, the cross-sectional area must be smaller further down.
What I don’t understand is how to confirm that the observed behavior matches the theory.
Thanks,
Rob
Take a picture and check the shape? Maybe you are asking something different…
Yeah, but you’d need a really tall column of water to start to see errors, wouldn’t you?
Thanks,
Rob
Yeah, that seems pretty easy. hang a sheet of graph paper behind the water stream, and put the camera far away (with a zoom lens) to minimize parallax error, and then you can measure the width of the water stream at various heights by seeing how its boundaries line up with the graph paper.
The relationship starts to fall apart when the speed of the water gets high enough for aerodynamics to start affecting things, or when the water has been falling for enough time to allow tiny imperfections to result in droplet formation due to surface tension, but if you’ve ever watched a real faucet trickling out water, you know that smooth, continuous flow can survive for several inches below the faucet outlet.
Depends how well you want to test it, as is the case for all scientific questions. However, I would suspect that the approximations taken in any simple derivation would start breaking down fairly visibly in a household faucet.
Fun fact: If you run a comb through your hair (so that it picks up a substantial static electric charge), and then bring the comb close to a very very narrow stream of falling water from a faucet, the stream will be deflected away from the comb.
Funner fact: This effect inspired Wilhelm Reich to build his cloudbuster. (Reich’s claim for how the cloudbuster worked: It drew deadly orgone radiation out of the atmosphere. The actual way a cloudbuster works: If you wait long enough, clouds will form and disperse on their own.)
That should be deflected towards the comb. The charge on the comb will attract the opposite charge to the water, and so will attract the water.
D’oh!
Proof positive that human memory (mine, in this case) fills in the gaps with made-up nonsense.
On the other hand, though, a powerful magnet will deflect the water away, since water is diamagnetic. That one’s harder to see, though, since few people have a sufficiently-powerful magnet lying around.
Feynman wasn’t considering the path of the stream of water - he was considering the shape of the stream as it comes out of a faucet. Here is a description by Feynman himself of the problem and how he solved it.
Yes. We all know that. And your link was already given, back in Post 2.