What is the equation that determines a dog's pursuit of a Frisbee?

[robert spicuzza]

Pursuit Problem:

A Frisbee is 40 ft north and 30 ft east of a dog.
The Frisbee is traveling north at 5 ft sec.

The dog can run at constant 10 ft/sec = SQRT( (Vdx)^2 + (Vdy)^2 )

Tan(angle)=Y(t)/X(t)

As the dog runs towards the Frisbee, the dog from “instinct” keeps the angle constant by adjusting his Vdx and Vdy closing velocities.

What is the equation of the curve that the dog travels along in catching the Frisbee? Picking an arbitrary time, say 5 seconds, what are the X and Y values of the equation. What are Vdx, and Vdy at 5 seconds?

Does anyone have a solution to this problem?


Other obvious questions:
Is the arc length of the pursuit equation a minimum, when the angle is kept constant? Is the time to catch the Frisbee also a minimum?

Thanks for any help on this. (It has been 30 years since I’ve solved any DE’s)

Dr Bob

 

[balakrishnan_v]

It will follow the equation
$$y=\frac{16+\sqrt{91}}{9} x$$
ie y=x*{16+sqrt(9)}/9 which is a straight line
Since the equation is a straight line,it is the optimal way of catching the Frisbee.So the time is minimum.

 

[leach]

I've tried to solve the problem, but the computations are rather complicated. I've found a reference about this problem in http://mathworld.wolfram.com/PursuitCurve.html" . The problem seems to be that the dog is twice as fast as the frisbee, and this prevents a fortunate simplification in the quadratic differential equation.

This doesn't mean, of course, that the problem can't be solved exactly, using other methods.

 

[balakrishnan_v]

@Leach
I think you misread the question.The problem is not to find the time of catch when the dog is always heading towards the frisbee.
The dog is chasing the frisbee such that the angle formed by the line joining frisbee and the dog with the east west line is a constant as the equation
atan(y(t)/x(t)) suggests.The solution for this is that the dog always travels in a straight line

 

[leach]

Yes, I think I misunderstood the question. Assuming that we make the hypothesis of constant angle, we find two possible trajectories for the dog, both straight lines. One of them is $$y=\frac{16+\sqrt{91}}{9} x$$, which you mentioned earlier, and the other is a divergent trajectory.