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Need help with spirals

  1. Mar 1, 2008 #1
    1. The problem statement, all variables and given/known data
    I'm participating in a quiz, where we need to complete a number of tasks over many months.
    This task involves math.

    Note: I am NOT asking for the solution, I'm just asking for a nudge, a formula, or something that I can use to calculate this, instead of just "hoping" that this technique works. I repeat, DO NOT solve this.

    You are on a platform. A creature wants to get you. You need to make it off the platform without being caught by the creature. It is to be assumed that the moment you are off (without being caught), you have succeeded.

    You on standing on the circular platform, of radius r. You move at speed v.
    Off the platform, is a creature. The creature cannot go onto the platform, however will do what it can to catch you. It will always take the most efficient path to get to you. The creature moves at speed 4v.

    2. Relevant equations
    Platform radius: r
    Your speed: v
    Creature's speed: 4v

    3. The attempt at a solution
    Using a straight-line method, and maximizing efficiency, you should stand in the center of the circle, with your back to the monster. Your goal is to run to the other side (A distance of r) before the creature goes around the semicircle to get you.

    v = d/t
    t = d/v

    t = r / v

    t = (1/2)*2pi*r / 4v
    t = (pi/4) r / v

    Since pi/4 is less than 1, regardless of r and v, the creature will get there first. This method fails.

    My next strategy is a spiral method. Same as before, start in the middle, have your back facing the creature. But, this time, as you start moving, continue adjusting your direction such that your back is ALWAYS facing the creature.

    To help visualize this, picture yourself in the center, facing North. The creature is at the South-most point of the platform. You start heading North. The creature will either go clockwise, or counterclockwise. If it goes clockwise, you start adjusting your direction to go more East.

    This will produce a spiral.

    The thing though, is will this get you to the edge of the platform? I don't know anything about spirals, but from plain logic, I see 3 outcomes:

    Creature is too fast: Not possible to get off the platform with this method. Creature won't catch you though, because as soon as it gets close enough, your back will face it, and you'll start heading into the center of the circle again.

    Time is tied: You reach the edge just as the creature reaches you. Technically counts as a failure.

    You succeed: Creature is too slow to catch you, so you make it off.
  2. jcsd
  3. Mar 1, 2008 #2
    I will entertain your spiral idea since I don't have a better idea myself, and will try my best not to be too specific. Things you know for certain are:

    -The magnitude of your velocity (v)
    -The magnitude of the creature's velocity (4v)
    -The path the creature will follow. You are at liberty to choose clockwise or counterclockwise, but its circular path may either be expressed as x(t)=r*cos(-90+wt),y(t)=r*sin(-90+wt) if you wish to consider the counter clockwise case, or x(t)=r*cos(-90-wt),y(t)=r*sin(-90-wt) if you wish to consider the clockwise case, where w is the angular velocity in radians per second (you know this also, what is it?). I consider east to be at 0 degrees so south would be -90 degrees where the creature starts in your case.

    From this you can find the directional velocity vector that your person wishes to use as well as the position. I hope I was clear (yet vague) enough.
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