Print

Please login or register.
1 Hour 1 Day 1 Week 1 Month Forever
Login with username, password and session length

[Grek]

You are located on an infinite plane of uniform density and in the possession of an infinite number of point-sized, perfectly accurate cannons which launch point-sized cannonballs at a fixed velocity of 100 m/s at arbitrary angles. All of the cannons will fire once (and only once) simultaneously upon a given signal. All collisions on this plane should be taken to be perfectly elastic. There is a constant wind of 5 m/s. Your mission, if you choose to accept it, is to define a set of axes to use for this problem, describe formulaically an arrangement of cannon positions, firing angles and trajectories that will allow you to stack, for a brief instant, an infinite number of point-sized cannonballs into as long of a line segment intersecting the origin that is perpendicular to the plane as possible and then to describe formulaically the locations which the cannonballs will impact the plane after having collided with eachother to form the tower and bounced back apart.

I considered posting this in the math help thread, but this is more for amusement than for any actual practical purpose.

[ed boy]

There is not enough information about the wind. You need to tell us what its vertical component is, and how much the cannonballs change momentum when acted upon by the wind.

[Akroma]

if the canonballs are point sized, their surface against the wind if infitisimal small.


so the wind does not matter




so, how strong is g on this plane, and how much do these canonballs weight ?

[Armok]

Umm, can't you just place the canons in a tower configuration, aiming exactly upwards?

[Bauglir]

Alternatively, in an infinitely tight spiral with a constant radius around the origin, with all of them pointed inward? Might not be any collisions, but you'll get a tower during the instant they all pass over the origin.

[Grek]

The wind is blowing 5 m/s in a single direction, parallel to the plane. Treat it as a constant 5 m/s^2 acceleration windwards.

g=9.81, same as on earth.

Placing the cannons in a tower formation isn't possible; cannons can only be placed on the ground. I should have specified that, admittedly. On a related note, cannons cannot be placed in the same place as other cannons. So, if you have one at [0,0], no other cannons can be placed there.

I'm not entirely sure how to interperate "an infinitely tight spiral with a constant radius" in a way that doesn't translate to "put them all in an infinitesimally small circle around the origin", which seems pretty questionable.

[squeakyReaper]

Direction of the wind doesn't matter in a global environment, since everything is uniformly flat and whatnot.  Actually, you didn't say that...  is the world flat?  Can we create elevations?  Do elevations block these cannon balls?  Would it be possible to just have an oval shape cannon formation and fire into a single point?

[Grek]

The world, being an infinite plane of uniform density, is indeed flat. Elevations cannot be constructed, at least not for this problem. Maybe for another?

Firing all of the cannons at a single point would not result in a tower, but rather a point followed by a massive ricochet in every direction.

[Bauglir]

I'm not entirely sure how to interperate "an infinitely tight spiral with a constant radius" in a way that doesn't translate to "put them all in an infinitesimally small circle around the origin", which seems pretty questionable.

I was having trouble expressing the idea, but picture a slinky for the basic idea, only it's made of an infinite number of points placed infinitely close together to form the line, and there's infinitely little space between the bands. It's basically a cylinder, but so constructed that no cannon is directly opposite another. I think. Doesn't matter, though, if vertical stacking isn't allowed.

[ChairmanPoo]

The real question is how many Maxwell's demons can dance on each cannonball?

[alway]

The number of points in which matter can occupy a space is not infinite. IIRC, trying to put 2 objects closer together than the Planck length will result in their waveforms interacting in such a way as to prevent one another from getting closer than the Planck length. The problem itself is flawed because things don't work that way; it's indeed similar to asking how many angels can dance on the head of a pin or how many Maxwell's demons can dance on each cannonball.

However, if it were refined to finite numbers, as is required by the laws of reality, you would end up, depending on whether or not you gave them mass, with a bunch of material which in general acted either like a very powerful light source or as a pressurized gas released into a vacuum (or a star or black hole would form, depending on how much mass and what the properties of particles were and how densely they were packed). Assuming of course these particles weren't doing anything funny like having nuclear force interactions and they weren't so dense as to form a black hole.

[Bauglir]

It would be more technically correct to refer to geometric points traveling along parabolic paths defined by their starting point and initial direction of travel, but "cannonball" seems fine, too.

[ed boy]

Direction of the wind doesn't matter in a global environment, since everything is uniformly flat and whatnot.

Nope. If the wind has a vertical component, then it will certainly matter. What's more, the wind could act in a non-linear fashion, such as in a spiral, which was why I had to ask about the wind direction.

[Bauglir]

How can the wind matter? Nothing of relevance has surface area for it to act upon, as mentioned earlier in the thread.

[Sowelu]

Yeah, spiral. Furthest in points straight up, farther ones aim lower and lower, with the farthest aiming at angle 0.  I guss you could put them in a line instead of a spiral, too, so let's say line.

Define the time it takes for the (0,0) 90-degree-up cannon to hit the highest point as T.  Define the distance of an angle-0 cannonball after time T as N.  Distribute all cannons along a line segment from (0, 0) to (N, 0).  Each cannonball fires so that it travels its x distance over time T.  Is that... angle acos(x/N)?