Get Off the Platform: Using Spirals and Maximum Efficiency

  • Thread starter Goldenwind
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In summary, this conversation discusses a task involving math in which the goal is to successfully make it off a circular platform without being caught by a creature. The key variables are the platform radius (r), your speed (v), and the creature's speed (4v). The conversation explores two potential strategies for achieving this goal, including standing in the center of the circle and using a spiral method. The conversation ends with a suggestion to consider the directional velocity vector and position in order to find a successful strategy.
  • #1
Goldenwind
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Homework Statement


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.

Homework Equations


Platform radius: r
Your speed: v
Creature's speed: 4v


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

You:
t = r / v

Creature:
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.
 
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  • #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.
 

What are spirals and where are they found?

Spirals are a type of geometric shape that consist of a continuously curved line that winds around a central point. They can be found in nature, such as in seashells or galaxies, and are also commonly used in art and design.

How can I create spirals?

Spirals can be created in various ways, depending on the desired medium. In mathematics, they can be created using equations such as the Archimedean spiral or logarithmic spiral. In art, they can be drawn or painted by hand, or using tools such as a compass or protractor. In digital design, there are also many software programs and apps that allow for the creation of spirals.

What are some real-life applications of spirals?

Spirals have many practical applications in fields such as architecture, engineering, and biology. In architecture, they can be used in the design of staircases, ramps, and other structures. In engineering, spirals are often used in machinery and technology, such as in the design of springs and gears. In biology, spirals can be found in the shapes of DNA molecules, plants, and various animal structures.

How can I use spirals in my scientific research?

Spirals can be used in scientific research in various ways. They can be used as a pattern for studying growth and development in plants and animals. They can also be used as a model for understanding natural phenomena, such as the formation of galaxies or weather patterns. In addition, spirals can be used in data visualization to represent complex information in a visually appealing and understandable way.

What are some interesting facts about spirals?

There are many interesting facts about spirals, including the fact that they are found in many unexpected places in nature, such as in the shape of a hurricane or the pattern of a pinecone. They are also considered to be a symbol of growth, evolution, and harmony. Additionally, spirals have been used in various cultures throughout history as a decorative motif and symbol of spiritual significance.

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