Curve of Pursuit

1. Jun 5, 2010

edGeykins

Suppose that a hawk, whose initial position is (a,0)=(3000,0) on the x-axis, spots a pigeon at (0,-1000) on the y-axis. Suppose that the pigeon flies at a constant speed of 60 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.

All the answers provided are correct, and I'm really close to the end I'm just stuck on one part.

The pigeon's position is given by the function (0, g(t)) where g(t)=60t-1000

The hawk's line of travel is tangent to the curve of pursuit, and is given by (1000+y-(x*p))/60
(where p=dy/dx)

The distance the hawk has flown is given by the integral ∫[x, 3000] √(1+p^2) which also equals 70t

The total distance can be considered: ∫[x, 3000] (1/70)√(1+p^2)

Next both sides of the equation are differentiated giving:

(-1/60)*(x*q) on the left and

(-1/70)(sqrt(1+p^2)) on the right

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

(p/sqrt(p^2+1)+1)/(sqrt(p^2+1)+p)=6/(7x)

Integrating we get

ln(p+sqrt(1+p^2)) on the left and

6/7(ln(x))+C on the right

solving for C we get =-6.535150622

and here's where the roadblock comes in. I cannot for the life of me solve for P.

a given hint was to "Exponentiate to get rid of the logarithm. Then isolate the square root on one side of the equation and square both sides" which is what I thought I did but I must be doing something wrong. I've been stuck on this for a good 2 hours and have gotten many answers but no success.

After solving for P, I have to solve for y, given that P=(dy/dx), then solve for C by plugging in the initial position of the hawk.

Any help would be really appreciated. I still have till Monday to have this done but it would be a huge load off my mind to finish it off by tomorrow.

Last edited: Jun 5, 2010
2. Jun 5, 2010

edGeykins

*update* After many hours of work and staring at the computer i was able to figure out the equation for P(= (x^(12/7)-e^(2*6.535150622))/(2 e^6.535150622 x^(6/7))) and from there i got y=((7 x^(13/7))/13-7 e^(2*6.535150622) x^(1/7))/(2 e^6.535150622)+C

plugging initial position of hawk into that (3000,0), C=6447.358594091420960 (For anyone in the future who has this problem use Wolfram Alpha to help with the extra decimals)

I'll keep this up in case anyone ever has a problem similar to this in the future

Last edited: Jun 5, 2010