I saw this problem posted on the internet somewhere and am intrigued how to solve it.(adsbygoogle = window.adsbygoogle || []).push({});

The problem is a fox chasing a hen with the following conditions:

1) The fox's starts at position x=0, y=10m (0,10)

2) The hen start's at position (0.0)

3) The fox runs at velocity 4m/s in a direction directly at the hen's current position.

4) The hen runs at a velocity of 3m/s along the x-axis.

The question is how long does it take the fox to catch the hen?

My approach to solve this starts with a position of the fox [tex]F(t)=F_x(t) \vec x+F_y(t) \vec y[/tex]

At this positions the velocity vector is [tex]VF(t)=VF_x(t) \vec x+VF_y(t) \vec y[/tex] where [tex]|VF(t)|=4[/tex]

The position of the Hen is 3t

Based on the position of the 2 animals the ratio [tex]\frac{VF_x(t)}{VF_y(t)}=\frac{3t-F_x(t)}{F_y(t)}[/tex]

Rearranging these equations I come up with the following 2 equations:

[tex]VF_x(t)=\frac{4(3t-F_x(t)}{\sqrt{(3t-F_x(t))^2-F_y(t)^2}}[/tex]

and

[tex]VF_y(t)=\frac{4F_y(t}{\sqrt{(3t-F_x(t))^2-F_y(t)^2}}[/tex]

From physics I know that [tex]v(t)=\frac{ds}{dt}[/tex] which I think I can break into: [tex]V_x(t)=\frac{dF_x(t)}{dt}[/tex] and [tex]V_y(t)=\frac{dF_y(t)}{dt}[/tex]

Plugging these 2 equations I come up with the following 2 equations which are a set of differential equations for Fx(t) and Fy(t):

[tex]\frac{dF_x(t)}{dt}=\frac{4(3t-F_x(t)}{\sqrt{(3t-F_x(t))^2-F_y(t)^2}}[/tex]

and

[tex]\frac{dF_y(t)}{dt}=\frac{4F_y(t}{\sqrt{(3t-F_x(t))^2-F_y(t)^2}}[/tex]

If I could solve for Fy(y) I would find t where Fy(t)=0

Are these equations solveable?

Secondly is my approach to solving correct?

This problem has got to be solveable since it seems simpler than the laws of planetary motion which involve accelaeration and 3 dimesions.

A coulple of observations I calculated are:

If the fox ran along the y-axis then along the x-axis the hen would be caught in 10 seconds so I know the aswer is less than 10seconds.

If the fox ran an angle of atan(3/7^0.5) the hen would be caught in 10/(7^0.5)=3.78seconds so I know the answer is greater than 3.78seconds.

If I run a recursive Excel spreadsheet with small delta t I calculate the answer to be approximately 5.715seconds.

I would use this number as a check of the solution.

Hope the post is legible. I am learning latex on the fly.

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# Difficult Differential Equations of Motion

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