bonanova

Izzy, if you want to just consider the geometry of this experiment, try writing all the simple relationships you know among the variables stated in the problem. Initial speed, launch angle, vx, vy, range, time. Then look for a relationship where only one of the variables is unknown. Plug that result into the other relationships and repeat.

It's more logical/mathematical than that.

Nice work Drexlin. Very Close.

The conclusion that they will fall requires assumptions. Assumptions that likely would make the original configuration fall.

Of what is today the meaning?

Hi jellyllama, and welcome to the Den.

I just met Jane. In many ways she's special, and we've become great friends. She tells it like it is, good and bad, without ever being critical. That's a gift I've come to cherish. She's also a crackerjack logician. So when I won a pair of tickets to the Annual Knights and Knaves Masked Ball at the Cosmopolitan Club, Jane was my date. I explained to her as we arrived that everyone inside was either a Knight [who would always tell the truth] or a Knave [who would always lie] and warned her that the conversations we'd hear might sound a bit bizarre as a result. We weren't disappointed. We introduced ourselves to a trio of club members, Dale, Gene and Lisa, who were conversing. Conversation 1. Lisa: I just figured it out: Gene is a Knave. Gene: Really? Well here's another scoop: You [Lisa] are the same as Dale. [both Knights or both Knaves.] Jane whispered, I'm not sure about the other two, but I know what Dale is. I smiled at my date's logical prowess and asked if she'd like some Chardonnay. She nodded. When moments later I returned with our drinks, Jane had a puzzle for me. While you were gone, she said, there were two more conversations, and I wrote them down. We can look at them later, tho; for now let's just dance and enjoy the music. It wasn't until Jane and I were relaxing at her place with a nightcap that she handed me her notes. Here, she said, see what you can do with these. Since we already knew a few things about our new friends, I removed all names and just used letters. She smiled as I read what she'd written. Conversation 2: A: B is a Knight. B: A is not a Knight. Conversation 3: [Not necessarily the same A and B as above.] A: B is a Knight. B: A is a Knave. It took me a moment, but I was finally able to reply. In each conversation, I said, I can identify one of the names. In one conversation, it's clear the named person was speaking with a Knave. In the other conversation, I'm really not sure. That's the best I can do. Jane smiled and said, You are, quite simply, amazing! What happened later that evening is material for a different forum. Can you determine what Jane concluded in Conversation 1? Can you reproduce my conclusions for 2 and 3? Good luck!

See

FWIW I've never seen a definition of tangent outside the plane. Tangent is one of the line-like properties [along with radius, diameter, secant, chord] of the circle. What distinguishes tangent from the others [except radius, I guess] is the single contact point. Well, if you go outside the plane, there's no need to make such a distinction. You can't have a radius, diameter, chord or secant outside the plane. So tangent needs a uniqueness statement [definition] only in the plane. In 3 dimensions there is only a single type of line that touches a circle. Although there are two types of lines outside the plane that don't touch a circle.

Bingo! http://brainden.com/forum/uploads/emoticons/default_thumbsup.gif' alt=':thumbsup:'> Can you imagine an intimate configuration for n=1?

Nice!

The topological proof that 4 is max isn't that hard to state. First, [my bad] the OP meant to restrict things to the plane, and the circles from intersecting. Well, if two circles are tangent, they can't also intersect, so that point is covered. OK. Begin by placing three circles [WOLOG they can be unit circles] so that they touch each other. This separates the plane external to the circles into an included area and the area beyond the circles. A mutually tangent fourth circle can be placed in either of these areas. But not both, because a circle in either area cannot touch one in the other without intersecting the original circles. And we can add only one in either area. Because, once added, it also isolates the tangency points, requiring intersections by any other circle.

Yes, that's the other one.

Right. Several years ago I chuckled at a New Yorker cartoon in which a young boy was tugging on the sleeve of his father, who was comfortably ensconced in his easy chair. The caption read: "Go ask your search engine!"

Simple problem . Imagine a 3-dimensional extension of the Thanks, now I don't have to draw it. Three spheres, of 3" radius lie on a table top, each sphere touching the other two. What is the largest sphere that fits in the space bounded by these four objects? That is, the largest sphere that lies on the table and between and under the other three spheres. . Harder problem . What is the greatest number of spheres that have all pairs tangent at distinct single points? How many configurations are there? Can you find a relationship among the radii?

You have it. Is that the only configuration?

Yes, the way the OP is [poorly] worded, that is admissible. The Hawaiian earring has a common point of tangency, admitting an infinite number of circles. The OP intended to require the pair-wise points of tangency be distinct. How many circles can have distinct pairwise tangent points? How many configurations exist? Can you find a relationship among the radii? Bonus question: can you relate this to the first, simple question?

Simple problem. . Two unit circles touch each other and a horizontal line segment. What is the radius r of the largest circle that fits in the space bounded by these objects? .Harder problem . What is the greatest number of circles that can have each pair of circles tangent at a single point? How many configurations are there? Can you find a relationship among the radii of the circles?