Need help writing a differential equation for this problem

[mondo11]

A Tortoise and a Hare make a bet.

The Tortoise bets that if he starts at the big boulder and runs due North, and the Hare starts at a distance L to the East of the big boulder and runs at the same speed right at the Tortoise, the Hare will never be able to tag him.

I need a step by step account of how to write a differential equation which could be used to analyze this problem.

Thank you.

 

[mondo11]

Here are pictures I drew for the problem stated





The problem expressed in pictures.

The first picture depicts the relationships of the boulder, the Hare and the Tortoise.

The second picture shows the relationships of the Hare (brown dot) and Tortoise (green dot) through ten ticks of the clock, where each tick is shown by a blue arrow.

You say: "I think that you will need to write a function that gives the position of the tortoise, followed by writing another function (in terms of the first) for the position of the hare."

The Tortoise is moving along the y axis at constant velocity, so

The position of the Tortoise at time t = y(t).

The position of the Hare in terms of the Tortoise = ycsc(t).

 

[mmm4444bot]

Wow, those are nice diagrams. Clearly, you understand the given scenario.

May I ask? Does this exercise come from a Differential Equations course? There are much easier ways to model this. For example, one could simply prove that the hypotenuse in any right triangle is always longer than either leg.

I'm not sure what your class is doing here.

The position of the Tortoise at time t = y(t)

The position of the Hare in terms of the Tortoise = ycsc(t)

For the purposes of discussion, let's agree that the animals' rate is measured in feet/second.

If we pick symbol y(t) to represent the distance-traveled by the tortoise, and we define y(t)=t, then we have implicitly stated that the tortoise's rate is 1 foot per second. I think that it would be better to write a general definition for function y (i.e., use a symbolic rate) because that would lead to a model that works for any speed.

Symbol L needs to come into play, as well.

I do not understand what you wrote for the hare; ycsc(t) is not a function notation that I am familiar with. How do you define ycsc(t)?

The animal positions are (x,y) coordinates, and we can certainly calculate distance from one point to another using the coordinates of the two points. That is, we could express the straight-line distance from the hare to the tortoise at any time t. If that expression can never reach zero, then we've proved that the hare can never reach the tortoise.

Anyway -- please confirm whether this exercise comes from a differential equations course; and, if it does, please post the exercise statement verbatim.

Cheers :cool:

 

[mondo11]

Here's the deal, abjectly speaking.

OK, I'll 'fess up.

You can probably tell, I am not a mathematician.

I do have a fairly recently (ten years or so) acquired passion for mathematics. Even though I am no good at it.

I did pretty well in calculus class, but that was by learning rules, not by understanding.

Now I want understanding.

I have always been bothered by the question of how to make a differential equation from scratch. (That's just one question, but a nagging one.)

So I said to myself, Mondo, I said, make up some simple problem and figure out how to make a differential equation from scratch.

Nothing like that kind of thing to expose your ignorance.

It's like learning a foreign language. Easy enough to learn to read it. Pretty easy to learn to converse. But try writing something original and you see you don't really know the language much at all.

The kind person who responded by saying "You have to try it yourself" is right, of course. But what I want here is not only "trying it myself" but watching myself try it, watching the thoughts that come into my head. A Socratic dialog, really, with awareness.

 

[mondo11]

Why did I pick this problem?

I wrote this problem down last year.

Since then, I am a bit more appreciative of how the Pythagorean theorem keeps on giving, appearing as a trope in so many different maths, and championed by category theory as an archetype of form.

I take your point of writing a more general expression for the Tortoise's position, one that can be used for any velocity, not just 1 foot/second. Wouldn't it be: y = Vt? V being velocity.

And thanks to Pythagoras, we can get the distance L as L² = h² - y2 .

Here is where my non-mathematical thought goes Bump in the night.

Aren't we supposing for some time t that h² = y² if the Hare tags the Tortoise? Or h² - y² = 0 or h² - V²t² = 0?

 

[mondo11]

Wait just a minute.

It occurs to me.

Maybe I am just recapitulating geometry, i.e. the Pythagorean theorem, when what I want is motion.

Maybe a better question, one more suited for differential equations, would be: How fast does the Hare have to run to catch the Tortoise? assuming that they are running along in our Newtonian world where we do not have to take into account relativistic corrections. So the differential equation needs a _________? What is this called in Mathspeak? An upper bound?

 

[mmm4444bot]

Hi -- I moved your thread out of the homework-tutoring area.

Well, off the top of my head, I don't know how much of a mathematician Socrates was, so I'm not sure how Socrates might have approached a deeper understanding of what a differential equation is.

The only way that I know of (to really understand differential equations), is to study them from a mathematician's point-of-view.

Differential equations come in many types, but the basic definition is: "any equation that contains a derivative".

You probably remember the following notation from calculus, yes?

dy/dx = 2x

It says that some unknown function (which has been named y) changes at a rate that includes x itself as a factor. In other words, in this example, y accelerates as x increases.

Well, that's a differential equation above. A very basic example. It's a differential equation because it contains a derivative (symbol dy/dx stands for "the rate at which y changes with respect to x").

The beauty of differential equations is that we can find out what a definition is for that unknown function, by solving the differential equation for y.

y = x^2

If you want to use differential equations to answer (or design) questions, then you need a firm foundation in the prerequisite subjects. (Unless, you desire to talk metaphysics!)

My suggestion is to start looking at Google hits, keeping what makes sense and skipping the rest. There are a whole lot of lessons and examples on the Internet.

Cheers :cool:

 

[mmm4444bot]

what I want here is not only "trying it myself" but watching myself try it, watching the thoughts that come into my head. A Socratic dialog, really, with awareness.

After my first read above, I wondered, "Is this guy an ex-hippy or something?". Sounds to me like maybe you ought to start a diary, if you want to capture all of that.

 

[mmm4444bot]

I am a bit more appreciative of how the Pythagorean theorem keeps on giving, appearing as a trope in so many different maths, and championed by category theory as an archetype of form.

I had to look up category theory (I briefly read the first bit at Wikipedia). Interesting. I don't know how the a^2+b^2=c^2 relationship figures, but the following question occurred to me when I read that "category theory allows many intricate and subtle mathematical results in [many significant areas of mathematics] to be stated, and proved, in a much simpler way".

Does one first need a thorough understanding of mathematical concepts, before one can simplify those concepts through category theory? Curious. :cool:

 

[mmm4444bot]

Wouldn't [a general function for the tortoise distance] be: y = Vt? V being velocity.

V = tortoise's rate

Yes, that would work for a constant-rate situation (i.e., tortoise always moves at the same speed, never goes faster or slower -- like in a Tandy Corp. video game).

And thanks to Pythagoras, we can get the distance L as L² = h² - y2 .

Here is where my non-mathematical thought goes Bump in the night.

Okay, it looks like you've got symbol h for length of hypotenuse? (At first, I was thinking "hare".)

When you say "get the distance L as", you're actually talking about expressing the quantity L in terms of h and y, I think. Yup -- that's how it's done.

Aren't we supposing for some time t that h² = y² if the Hare tags the Tortoise?

Because both h and y are greater than zero, we may simplify h^2=y^2 to h=y.

Yes, if the hare were to catch up, h would equal y. But that's impossible! There can be no triangle, if h equals y. That's the basis for a proof; to show that h can ever equal y.

I understand that the original problem as posed has gone somewhat to the wayside (not really a diffeq application), so I won't elaborate on how the proof that I had in mind would go, but it uses the distance formula with coordinates of the animals locations at time t.

 

[mmm4444bot]

Maybe a better question, one more suited for differential equations, would be: How fast does the Hare have to run to catch the Tortoise?

Faster than the tortoise, that's for sure!

How about: where do they meet, if the hare runs 20 times faster than the tortoise?

By the way, do you know that there is a method of proving the Pythagorean Theorem using a differential equation? I'm sure that it can be found online, if you're interested.

 

[HallsofIvy]

We can choose our units so that the tortoise's speed is 1. Suppose that the hare moves at speed v. Then at time t, the tortoise is at (0, t) and we can take the hare to be at (x, y). The hare moves directly toward the tortoise so it will move in the direction of the vector <-x(t), t- y(t)>. That has length \(\displaystyle \sqrt{x^2+ (t- y)^2}\) so the hare's velocity vector is \(\displaystyle \left<-\frac{vx}{\sqrt{x^2+ (t- y)^2}}, \frac{v(t- y(t))}{\sqrt{x^2+ (t-y)^2}}\right>\).

That is, the motion is given by \(\displaystyle \frac{dx}{dt}= -\frac{vx}{\sqrt{x^2+ (t- y)^2}}\), \(\displaystyle \frac{dy}{dt}= \frac{v(t- y)}{\sqrt{x^2+ (t- y)^2}}\).

 

[mondo11]

Reply

Faster than the tortoise, that's for sure!

How about: where do they meet, if the hare runs 20 times faster than the tortoise?

By the way, do you know that there is a method of proving the Pythagorean Theorem using a differential equation? I'm sure that it can be found online, if you're interested.

I read there are thousands of proofs of the theorem. I will look this up as it sounds enjoyable.

 

[mondo11]

I'm away until tomorrow

Thank you for the conversation. I am away until tomorrow, then I hope to get back to this. Please don't go away.

 

[mondo11]

My day job

After my first read above, I wondered, "Is this guy an ex-hippy or something?". View attachment 2310 Sounds to me like maybe you ought to start a diary, if you want to capture all of that.

I do neuroscience research evenings and week ends. My day job is as a clinical psychiatrist with severely and chronically mentally ill people. Thus. I have an immediate and pressing interest in inner experience - there are times when my life depends on it! Not to over state the case, or anything. I am also interested in "embodied cognition" viz the excellent book by Lakeoff and Nunez, WHERE DOES MATHEMATICS COME FROM?

My neuroscience research is under the aegis of theoretical neurobiology, particularly theories of global brain functioning and the structure of consciousnesses. I have selected 12 mathematical formalisms currently in play by different research groups, and I am looking through them one by one to learn them (or at least get a feel for them) on the road to trying to understand brain functioning better. If you have the time, check out westillknowsquataboutthebrain.com.

One really nice part of my research is getting to look in more depth at problems left unanswered from my university days. My hubris is such that I now feel "no question is too stupid if I ask it."

 

[mondo11]

I am ill posed. But another question arises.

Having perused the replies in this thread, the Hare and the Tortoise had the following conversation.

Hare: Well, I feel I must tell you that I cannot enter the wager you have suggested.

Tortoise: And why is that, pray tell?

Hare: Because, dear Tortoise, you have failed to specify the initial conditions under which the wager is to take place. Surely you can see that no matter how far or near East of the boulder I start, there are three sets of initial conditions. For two of them, I shall never tag you and hence shall always lose the wager. For one of them, I shall always tag you, and hence shall always win the wager. We can know with certainty without even running the race, what the outcome would be.

Tortoise: Pray be so kind as to settle my mind with an explanation.

Hare: It all has to do with the Theorem of Pythagoras, which states that my route will always be longer than yours. If my pace is slower than or equal to yours, I shall always lose. But if my pace is faster, I shall always win. Your wager, you see, is ill posed.

So I have realized something that I knew all along. A problem must be well posed. Before a math can be applied to a real-world adventure, that adventure must be somehow commensurate with the math chosen. This is not to minimize what I have learned. A realization is always a step-forward - a "making real" of something we may have held only dimly in mind, if at all.

What I want is to start from scratch and construct a differential equation. Now that I know that, there seem to be a lot of places to look at how that is done.

But another question comes out of all this.

Given some problem (I leave this purposely vague), how do you decide which kind of mathematics to try on it? Is this just a question of trial and error? Is there a technical term for this? Is this an area of study?