Calculus/Dog Curve

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Expert: Alon Mandes - 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
Suppose that the owner's velocity is a & the dog's velocity is b.
At a given point M(x,y) on the curve y=f(x) we have :
f'(x)=y'=(y-at)/x or xy'=y-at. Note that both x & y are function of time. & at is the point where the tangent line in the point M intersect with y-axis.
We also know that ds=-dx√(1+y'2) & ds/dt=b. ("where ds is the differential element of the curve at point M. & the - sign is because the tangent line has negative slope").
That gives us dx/dt=b/√(1+y'2).
Let's calculate dy/dt  : dy/dt=(dy/dx)*(dx/dt)=y'*dx/dt.
Let's calculate dy'/dt : dy'/dt=(dy'/dx)*(dx/dt)=y''*dx/dt
Now. let's take the 1st equation xy'=y-at & derive it with respect
to t :
d/dt {xy'}=d/dt{y-at}
y'*[dx/dt]+x*dy'/dt=dy/dt-a
y'*[dx/dt]+x*y''*[dx/dt]=y'*[dx/dt]-a
xy''*[dx/dt]=-a
xy''=-a/b/√(1+y'2)
xy''/√(1+y'2)=a/b & this is the equation of the Curve.
Solving this differential equation involves substitute method x=sh(u)
("sh is sin hyperbolic"). The curve graph will be :
If a=b then y=f(x)=x^2/4 - Ln(√x) - 1/2
If b=2a then y=f(x)=(x√x)/3 - √x + 2/3

Alon.

Alon Mandes

Expertise

Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .

Experience

1. I'm a team member of mathnerds (math site for answering questions) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

Organizations
Hi-Tech company : GSM4VOIP ; job possition : Algorythm developer.

Education/Credentials
M.A in Mathematics & Bs.c in Electronics.

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