I'm assuming that it's the object's trajectory, and not the object itself, that is bending. Think of a toy car with wheels that don't turn. If you roll the car from a tile floor onto a carpet, at an oblique angle to the boundary, the car will turn. This is because the wheels on one side are traveling at a different speed than the wheels on the other side. You might get the same effect by sliding a piece of two-by-four across the tile floor. It'll tend to spin around when it hits the carpet. A ball, on the other hand, has only one contact point and so would be less likely to turn as it crosses the boundary. All of the above assumes that there is no lip or crack or other place that causes bouncing at the interface. In the case of tossing stuff into a pool, the trajectory will be altered by the water resistance. None of this is the same as Snell's Law, in which ALL of the trajectory change occurs at the interface between two media. For a physical object traveling in a medium, the trajectory is continuously altered by the medium, i.e. friction, air resistance, or water resistance.
From: Forum for Physics Educators on behalf of Carl E. Mungan
Sent: Thu 8/12/2004 11:41 AM
Subject: bending of object thrown into pool
If a macroscopic object (say an arrow or a golf ball) is thrown into
a pool at an oblique angle, what will be the relation between the
incident and transmitted angles?
I was perusing the book "What Einstein told his Barber" in which the
author (a retired chemist) claims Snell's law would be obeyed. I'm
finding it hard to see this. It seems to me that one should start by
resolving the incident velocity into normal and tangential
components. The transmitted normal component would presumably depend
on the elastic surface tension, while the tangential component would
only respond to the viscosity. Once it's in the water, the drag force
is of course oppositely directed to the velocity and so does not bend
the trajectory, although I wonder about the situation when the arrow
is half in and half out of the water?
I have a dim memory that Newton once argued from his corpuscular view
that light particles should bend *opposite* to the wave prediction of
Snell's law. If someone remembers why that should be so, I'd be
grateful for a primer on the subject.
In any case, the standard derivation of Snell's law (continuity of
the electric field crests and troughs, together with an
index-dependent change in velocity) doesn't seem to me to be readily
adaptable to particles. On the other hand, what about electron
microscopy say? How does the size of the de Broglie wavelength figure
into Snell's law? There's the well-known question about the fastest
path to get from point A on the beach to get to point B in the water,
which is sometimes compared to Feynman's path-interference model of
light propagation: does this mean that coherence is also a factor?
If somebody on the list has a pool, I would be very interested if
they could actually try launching various objects into the pool and
report back on their observations. Carl
ps: Theoretically, it might be best to imagine the object to be
neutrally buoyant, because we're not interested in the effects of
gravity and buoyancy. Experimentally I don't think these matter too
much for dense objects over short distances.
pps: I just tried rolling a ball from a smooth book onto a rug at the
same height and see no obvious bending, although I'll admit the
results appear to depend on whether there's any remaining lip or
crack between them.
Carl E. Mungan, Asst. Prof. of Physics 410-293-6680 (O) -3729 (F)
U.S. Naval Academy, Stop 9C, Annapolis, MD 21402-5040 mailto:email@example.com://usna.edu/Users/physics/mungan/