Differential Equation of a hawk

[stosw]

Homework Statement

Suppose that a hawk, whose initial position is (a,0)=(3000,0) on the x-axis, spots a pigeon at (0,-2000) on the y-axis. Suppose that the pigeon flies at a constant speed of 50 ft/sec in the direction of the y-axis (oblivious to the hawk), while the hawk flies at a constant speed of 90 ft/sec, always in the direction of the pigeon.

The fact that the hawk is always headed in the direction of the pigeon means that the line PQ is tangent to the pursuit curve y=f(x). This tells us that (dy/dx)=h(x,y,t) where h(x,y,t) = ?

The Attempt at a Solution

I found the equation for the pigeon to be

g(t) = -2000 + 50t

I have no idea how to find an equation for the hawk. Any tips/hints/suggestions would be great.

 

[ideasrule]

This question is easier than you might think. Since the hawk is flying towards the pigeon, the tangent line of its path always intersects the pigeon's position. If you write out the equation of the tangent line, it will be obvious what dy/dx is.

 

[HallsofIvy]

Homework Statement

Suppose that a hawk, whose initial position is (a,0)=(3000,0) on the x-axis, spots a pigeon at (0,-2000) on the y-axis. Suppose that the pigeon flies at a constant speed of 50 ft/sec in the direction of the y-axis (oblivious to the hawk), while the hawk flies at a constant speed of 90 ft/sec, always in the direction of the pigeon.

The fact that the hawk is always headed in the direction of the pigeon means that the line PQ is tangent to the pursuit curve y=f(x). This tells us that (dy/dx)=h(x,y,t) where h(x,y,t) = ?

The Attempt at a Solution

I found the equation for the pigeon to be

g(t) = -2000 + 50t

I have no idea how to find an equation for the hawk. Any tips/hints/suggestions would be great.

Are you sure you have stated the problem correctly? As it is, as ideasrule said, it is a trivial problem. Perhaps the pigeon is not flying "in the direction of the y-axis" but "parallel to the y-axis"?

Also you said "I found the equation for the pigeon to be g(t) = -2000 + 50t". Well, pigeons don't have equations! If you meant that to be the equation for the position of the pigeon, shouldn't that have both x and y components? Which does that equation give?

 

[mikhailpavel]

i am also confused regarding the question...can anyone post the detailed solution.

 

[stosw]

Are you sure you have stated the problem correctly? As it is, as ideasrule said, it is a trivial problem. Perhaps the pigeon is not flying "in the direction of the y-axis" but "parallel to the y-axis"?

Also you said "I found the equation for the pigeon to be g(t) = -2000 + 50t". Well, pigeons don't have equations! If you meant that to be the equation for the position of the pigeon, shouldn't that have both x and y components? Which does that equation give?

I'm almost positive the pigeon is moving only in the direction of the y-axis.

You're right they don't have equations, lol. The pigeon's position Q=(0,g(t)) where i found g(t) = -2000 + 50t.

Could I then say something like the hawk's position P = ( j(t), k(t) ) where

j(t) = 3000 - 90t and k(t) = 90t

(Based on the starting position of the hawk on the x-axis)?

If I can do that, I'm not seeing how to get from here to the equation for h(x,y,t) that involves all three of those variables.

Due to it being the tangent line and me having two points P and Q, could I used y = mx +b? If so, I'm stuck again at how to get to h(x,y,t).