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A Weekend of Questions in 1 Post – Go Wild ;-)

Posted by Karl Sharman on September 17, 2010 – 7:05 am

1. The answer to this question is almost in the question itself! You have a ham sandwich with melted cheese on toast. With a single, perfect slice traveling along a two-dimensional plane, is it always possible to divide the sandwich so that the two halves have equal parts ham, cheese and toast? You may not assume an ideal sandwich- the ingredients may not necessarily be lying perfectly flat.
2. What is the fewest number of knights on a chess board so that every square is either occupied by, or under attack by, at least one knight?
3. What is the mass of Earth’s gravitational field?
4. An assassin with a high-powered rifle shoots you in the foot from 50 feet away. The bullet travels at 1300 feet per second. You and the assassin are at sea level. What will be the first evidence to you of the attack?
5. What do the following have in common? “Johnny B. Goode”, “Rite of Spring”, “Beethoven’s 5th”, “Dark Was the Night” and “Melancholy Blues” (Yes, we know they’re all music, thank you.)
6. Fly due south from the capital of Florida until you are at the same latitude as the capital of Brazil. What country or body of water is beneath you?

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  1. 1. John24 Said:

    1. Not “always” possible
    2. 12 knights
    3. No mass
    4. The sound. Even though the bullet travels faster than the sound, the pain from your foot to your head will take enough time that you will hear the sound just before you feel the pain. Pain travels at about 10 ft per second.
    5. not a clue…yet
    6. Cook Islands or very close if using the Magnetic South Pole.

  2. 2. mightbewrong Said:

    i’m going to skip some of these.
    #4) if there were muzzle flash, the light would reach the eye in 5.08×10^-8 sec, but with human reaction time (to light = 0.19 sec), it will register in about 0.19 seconds.
    the sound will travel in 0.044 sec, but the brain has to process it, so add 0.16 seconds for that, and you get 0.20 sec.
    the bullet gets there in 0.038 sec, but the pain will travel in approx. 0.361 sec for a total of 0.399 sec.
    SO….muzzle flash first, followed by the sound close after, and finally the throbbing pain of a missing foot.
    #6) South Pacific Ocean.

  3. 3. mightbewrong Said:

    #6) Tallahassee, FL (30.4,-84.3)
    Brasilia, Brazil (-15.8,-47.9)
    (-15.8,-84.3) = smack dab in the middle of the Pacific. nothing around.

  4. 4. mightbewrong Said:

    #1) it says “halves”, so that must mean they are two equal parts that make a whole. otherwise you hould have to call them parts or segments or something different.
    if not for the word “halves”, we would need to know if it means by count (2 parts toast, 1 part cheese, 20 parts ham; i like meat), by mass, by volume, and if the slice plane is even perpendicular to any of the contents. in any case, i would say no, not “always” is it possible. however, my simple answer to this questions is “YES” because of the word “halves”.

  5. 5. Chris Said:

    1. I can’t think of a case where it isn’t true.

    3. Greater than 0. I’ll do some calculations later. But it’s because a gravitational field has an energy density and E = mc².

  6. 6. Chris Said:

    hi mightbewrong. I’m sure Karl means exact halves. Each half might have a different shape. I’m pretty sure that it doesn’t matter if he means by volume or by mass.

  7. 7. Chris Said:

    Hmmm. The whole field above the surface has a mass of about 950 tonnes.

    I see little point in showing how I calculated that.

  8. 8. Chris Said:

    … perhaps the should be -950 tonnes (i.e. a negative mass).

  9. 9. Chris Said:

    I’m pretty sure I goofed the numbers. 950 tonnes is way too small. I now think that the mass outside the Earth is 2.1*10^12 tonnes and the mass inside is .42*10^12 tonnes (for the latter I’ve assumed a uniform density inside the Earth).

  10. 10. Chris Said:

    The energy density of a gravitational field is U = g²/(8πG), where g is the acceleration due to gravity (at the point of interest) and G is the universal gravitational constant. The mass density of a gravitational field is therefore U/c².

    Outside the Earth’s surface, g = GM/r², where M is the mass of the Earth and r is the distance from the centre (r > R, where R = the Earth’s radius).

    Assuming a uniform density for the Earth, inside the Earth g = GMr/R³.

    Now simply integrate those (with respect to volume), divide by c² and you have the mass of the field.

    As to what contibutes to the Earth’s mass; that seems to be a matter of definition, how you measured the mass and from where you measured the mass.

  11. 11. Chris Said:

    oops, I meant integrate the U fields (I seem to have implied integrating the g fields).

  12. 12. Karl Sharman Said:

    The Answers:-

    1. Yes. In fact there’s a mathematical theorem which literally states this. It’s called the Ham Sandwich Theorem (no, really).
    2. 12 – I have seen a diagram depicting this, but I have no idea how you would actually prove it mathematically
    3. -3 trillion tons. The Earth’s gravitational field has energy, which according to relativity, has mass. You can read about it at Scientific American or at the home page of Gravity Probe B
    4. If you happened to be looking straight at him, the flash from his rifle would reach you almost instantly. Assume for a moment that the he’s a competent assassin who hid himself well (and for some reason was aiming for your foot). The speed of sound is approximately 1129 feet per second at sea level. The sound will reach your ears in .044 seconds. The bullet will hit your foot in .038 seconds. So much for dodging the bullet on this one. But we asked which would be the first evidence to you- and that would be the sound. Because once the bullet hits your foot, the pain impulse must travel your myelinated A-fibers to your brain at 330 feet per second. Assuming you’re five feet tall, that’s an extra .015 seconds onto the trip
    5. They were all on the first album to go “trans-solar”, on Voyager.
    6. The Pacific Ocean. Most of us accustomed to Mercator projection maps think of North America being more or less lined up directly above South America, but look on a globe and see that almost all of South America is east of North America.

  13. 13. John24 Said:

    I do buy the fact that 3 spheres placed randomly in space there must exist a plane which produces equal portions of each sphere. This is regardless of the size of spheres used.

    I cannot buy this theorem as proof. What if you have a sandwich with all of the ham lying on one half of the sandwich with the cheese lying in one quarter of the sandwich, either on the ham half or on the other half. Where is the plane that would create equal portions of bread, ham & cheese?

    This theorem is not supposed to create a sandwich in the ordinary sense but rather 3 objects regardless of size and shape can be bisected perfectly by a single plane. Placing objects within each other would ruin this theorem IMHO.

  14. 14. Chris Said:

    Hi John24. Each object, regardless of its shape and size, has a centre (the centre may not actuall be inside the object). Any plane going through that centre must bisect it. We have three centres, and so can find a plane that bisects all three objects.

    It’s not difficult to see that this is true regardless of whether we are talking about bisecting the volumes or the masses of the objects.

  15. 15. Ultra Man! Said:

    1. The answers is no. It is not always possible to cut it perfectly in half because it just says “cheese”, not any specific type of cheese. Therefore, you can have 2 pieces of cheese, like cheddar and parmesean, and insteading of having 3 objects that need to be cut in half, you have 4, which is only possible if they are all on the same plane.

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