Re: Fun problem
- From: "Jay R. Yablon" <jyablon@xxxxxxxxxxxx>
- Date: Sat, 29 Apr 2006 06:43:28 +0000 (UTC)
The below was never posted, sent about 4-5 days ago. Jay.
"Igor Khavkine" <igor.kh@xxxxxxxxx> wrote in message
news:1145745718.411724.269370@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
I recently ran into a fun problem from an old Russian physics olympiad.
I found the solution a little surprising and intriguing. The problem
requires no more than high school classical mechanics and some
ingenuity. Here it is for your amusement.
Consider a point particle sliding on a flat table (ignore friction).
The table has a cylindrical hole of finite depth (vertical walls, flat
bottom). The particle can approach the hole with different velocities
and with different impact parameters (the particle's motion need not be
directed toward the center of the hole). As the particle falls into the
hole, it starts bouncing off the walls and the bottom (assume elastic
collisions). Sometimes it gets stuck in the hole forever, sometimes it
escapes (bounces out). Determine the relation between the depth of the
hole, its radius, the particle's initial velocity, and impact parameter
necessary for the particle to escape after it falls in.
Enjoy!
Igor
Initial impression of how to approach this:
Start by looking at the case where the particle goes straight to the
center of the hole (direct impact parameter -- great putt!). If the
hole has a radius r, the particle will jump back out if its velocity is
such that it strikes the bottom of the hole at the exact center, because
its upward path after hitting the bottom will precisely reflect the
downward path due to the perfect elasticity assumption. It will bounce
out at the opposite side of its entry, and continue forward.
The next way for it to escape is to hit 1/3 of the way across. It will
then bounce up and reach escape height 2/3 of the way across, and hit
bottom again the whole way across. Then, it will precisely reverse
trajectory and come out the same way it came in, and continue back
toward the golfer. (Sounds like my putts.)
At 1/4, it will escape after a second bounce at 3/4, and move forward on
escape. At 1/5. escape but backwards. 1/6, escape but forward, etc.
1/n where n is even is forward escape. 1/n where n is odd is backward
escape.
Now, let's look at 2/3. Hits bottom at 2/3, 6/3, 10/3, 14/3 etc. Hits
top at 4/3, 8/3, 12/3. 12/3=4 is the answer. Goes back and forth twice
(four traversals), then escapes backwards.
Next, 3/4. Bottom at 3/4, 9/4, 15/4, 21/4. Top at 6/4, 12/4, Forward
escape after three traversals.
Next, 4/5. Top (that what counts) at 8/5, 16/5, 24/5, 30/5. Bingo.
Six traversals, backward escape.
Next, 3/5. Top at 6/5, 12/5, 18/5, 24/5, 30/5. Again, six traversals,
backward escape.
Next, 2/5. Top at 4/5, 8/5, 12/5, 16/5, 20/5. Backward escape, four
traversals. We did 1/5 before.
Next, 5/6. Top at 10/6, 20/6, 30/6=5 traversal, forward escape.
General rule: escapes if velocity, radius and depth (and the
gravitational acceleration) are such that it strikes the bottom m/n of
the way across the hole, where m and n are integers. If n is even it is
forward escape. If n is odd it is backward escape.
The impact parameter at first thought does not change anything: it still
needs top strike m/n of the way across the hole on whatever path it
takes toward the hole. This means that for a less than square impact,
the velocity will need to be reduced proportionally to the length of
hole over which it travels. BUT, on second thought: when it hits the
hole wall, it will not bounce straight back any more. It will go at an
angle. I suspect that you will then also need to consider paths which
are triangles, squares, pentagons, etc. That how I'd lay this out. The
quantization m/n will carry over to the angles of incidence, which will
have similar quantum constraints. That is step 2 of this problem. The
interesting result here will be the angles at which it bounces out.
One should also consider the extreme case of an exact tangential graze.
Then, as I think was pointed out earlier, it will spiral in, hit bottom,
then bounce back out. The exit angle will depend on depth, radius, and
velocity.
After all of this is laid out, then one reverse engineers the math to
get the formula involving the "depth of the hole, its radius, the
particle's initial velocity, and impact parameter," assuming, e.g. a
sea-level gravitational acceleration on earth at the equator, or some
such thing.
Jay.
_____________________________
Jay R. Yablon
Email: jyablon@xxxxxxxxxxxx
.
- References:
- Fun problem
- From: Igor Khavkine
- Fun problem
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