Calculus/Dog Curve

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Expert: - 11/10/2008


Question

Dog Curve
Hi James;

Here is a problem I can't solve:
A dog is standing in "B" point on the "X" axis. It's owner is standing in the center(cross of X and Y axis).The owner begins to run in the Y direction and the dog wants to follow the owner in a form that his face(velocity)direction is always to the owner.
The dog reaches the owner somewhere on the "Y" axis. How can we drive the graph for the dog's way?
The curve made by the dog is called the "Dog Curve"(courbe chien) in physics and calculus.
Can you help?
I attached the graph to the question.

Looking Forward To Your Quick Response.

Mohammad

Answer
Category:  Calculus
Private:  No
Subject:  Dog Curve
Question:  

Dog Curve

Hi James; <<<< Huh?

Here is a problem I can't solve:
A dog is standing in "B" point on the "X" axis. It's owner is standing in the center(cross of X and Y axis).The owner begins to run in the Y direction and the dog wants to follow the owner in a form that his face(velocity)direction is always to the owner.
The dog reaches the owner somewhere on the "Y" axis. How can we drive the graph for the dog's way?
The curve made by the dog is called the "Dog Curve"(courbe chien) in physics and calculus.
Can you help?
I attached the graph to the question.

Looking Forward To Your Quick Response.

Mohammad
................................
Hi, Mo,

This is as far as I got:

The man is at O, and the dog is at  (b,0)

The man runs at a speed of r.
His position is  M(0,rt)

The dog runs at  ds/dt = v

If the dog is at D(x,y), then the slope of the line from D to M is

m(t) = (y - rt)/(x - 0)

m(t) = (y - rt)/x

And that is  m(t) = (dy/dt)/(dx/dt) = y*/x* [x-dot over y-dot]

But  v = ds/dt = sqrt(x*^2 + y*^2)

   v^2 = x*^2 + y*^2

y* = m(t) x*

y* = (y - v0t) x*

y*^2 = (y - v0t)^2 x*^2

v^2 = (y - v0t)^2 x*^2 + x*^2

Now that gets messy.  Here are a few references I was able to find: (it's called a PURSUIT problem)


http://www.2dcurves.com/power/powerp.html
http://curvebank.calstatela.edu/pursuit/pursuit.htm
http://www.physicsforums.com/archive/index.php/t-114457.html

Paul Klarreich

Expertise

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

Experience

I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

Education/Credentials
(See above.)

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