Curve of Pursuit (Pigeon and Hawk) (Detailed)

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In summary, the hawk's flight path is tangent to the curve of pursuit and is given by p=(-2000+y-(x*p))/40. The total distance the hawk has flown is given by the integral ∫[x, 7000] (1/70)√(1+p^2). The time and place where the hawk will catch the pigeon can be found by solving for y and integrating.
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whstrack2010
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Homework Statement



Suppose that a hawk, whose initial position is (a,0)=(7000,0) on the x-axis, spots a pigeon at (0, 2000) on the y-axis. Suppose that the pigeon flies at a constant speed of 40 ft/sec in the direction of the y-axis (oblivious to the hawk), while the hawk flies at a constant speed of 70 ft/sec, always in the direction of the pigeon.

The problem is to find an equation for the flight path of the hawk (the curve of pursuit) and to find the time and place where the hawk will catch the pigeon. Assume that in this problem all distances are measured in feet and all times measured in seconds. Leave out all dimensions from your answers.

The Attempt at a Solution



All of this information is what I have done, and I know it to be correct.

The pigeon's position is given by the function (0, g(t)) where g(t)=2000+40t

The hawk's line of travel is tangent to the curve of pursuit, and is given by p=(-2000+y-(x*p))/40 where p=dy/dx

The distance the hawk has flown is given by the integral ∫[x, 7000] √(1+p^2) which also equals 70t.

So, the total distance can be considered: ∫[x, 7000] (1/70)√(1+p^2)

Now, both sides of the equation (p=(-2000+y-(x*p))/40 and ∫[x, 7000] (1/70)√(1+p^2)) are differentiated to get rid of the integral.

On the left, we get (-1/40)*(x*q) where q=dp/dx
On the right, we get (-1/70)(sqrt(1+p^2)) by the Fundamental Theorem of Calculus.

This is now a separable equation with the variables p and x. It can be equated to:

1/sqrt(1+p^2) = (4/7)1/x

Integrating the left, we get ln(p+sqrt(1+p^2))

Integrating the right, we get 4/7(ln(x))+C



Now, here's where I'm stuck. I need to find C (it says to recall that p=dy/dx=tangent line from the beginning) , and then I think I use that to solve for p. Then that would be integrated to find y, which would have another C that I would need to find. And with that, I could then find the time and point where the pigeon is caught. Anyone have any suggestions on how to do this?
 
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  • #2
I think you meant to say t=(-2000+y-(x*p))/40, not p=(-2000+y-(x*p))/40. I'll assume the rest of that is ok. To find the C you need the initial value for p and the initial value for x, right? The initial value of the slope dy/dx is the slope of the hypotenuse of the triangle made by the initial positions and the origin, isn't it?
 
  • #3
yes, sorry about the mistake. i think that would make sense though.
 
  • #4
however, if you solve for p there as √(a^2+b^2), and you plug in, but i don't know the initial value of x.
 
  • #5
whstrack2010 said:
however, if you solve for p there as √(a^2+b^2), and you plug in, but i don't know the initial value of x.

p is a slope, not a length. And it sure looks to me like the initial value of x is 7000.
 
  • #6
well then how am i to solve for p? i know that it equals (40t+2000-y)/-x. But t is a function of x y and p. I thought that p would be (D*x^4/7-(1/D*x^4/7))/2 where D is -b+√(a^2+b^2)/(a^1+R), which I recently found as a posted solution to a question very similar to this just with different numbers, but that doesn't seem to be helping me out.
 
  • #7
Ok, use p=(40t+2000-y)/x. Initial t=0. Initial x=7000, initial y=0. x and y are the coordinates of the hawks position.
 
  • #8
Ah, I got it. Yeah, by using p=-2/7 and x=7000, I got C=-5.3412

So now I have to solve for p, and then integrate. alright.
 

Related to Curve of Pursuit (Pigeon and Hawk) (Detailed)

1. What is the Curve of Pursuit (Pigeon and Hawk)?

The Curve of Pursuit (Pigeon and Hawk) is a mathematical model that describes the path a pigeon takes when being chased by a hawk in pursuit.

2. How does the Curve of Pursuit (Pigeon and Hawk) work?

The model takes into account the speed and agility of both the pigeon and the hawk, as well as the distance between them. It calculates the optimal path for the pigeon to take in order to evade the hawk's pursuit.

3. What factors affect the Curve of Pursuit (Pigeon and Hawk)?

The Curve of Pursuit (Pigeon and Hawk) is affected by the speed and agility of both the pigeon and the hawk, as well as the distance between them. It is also influenced by any obstacles or terrain that may be present.

4. Can the Curve of Pursuit (Pigeon and Hawk) be applied to other situations?

Yes, the model can be applied to other situations where one object is pursuing another, such as predator-prey relationships in nature or in sports like soccer or football.

5. How is the Curve of Pursuit (Pigeon and Hawk) useful?

The Curve of Pursuit (Pigeon and Hawk) can help us understand the behavior of animals in pursuit situations and can also be used in designing strategies for evading pursuit in various scenarios.

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