Difficult Differential Equations of Motion

In summary: However, I don't think I would have been able to solve them analytically. Any thoughts?In summary, the conversation discusses a problem involving a fox chasing a hen with specific conditions. The approach to solving the problem involves deriving equations for the positions and velocities of both animals. The equations can be simplified by using the derivative dy/dx to represent the direction of the fox's travel. The solution to the problem involves solving a set of coupled differential equations, which can be done using numerical methods.
  • #1
RFurball
3
0
I saw this problem posted on the internet somewhere and am intrigued how to solve it.
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.
 
Physics news on Phys.org
  • #2
Hint: Denote Fx simply as x and Fy as y.

What is dy/dx?
 
  • #3
This problem (or equivalent) can be found in many differential equations textbooks:

Flying a plane in a crosswind, always heading straight toward the destination.

Rowing a boat across a river, always heading straight toward the destination.

I first heard it as a farmer chasing a pig.

Here is a solution I wrote up a few years ago

http://www.math.ohio-state.edu/~edgar/H521_02/HW8s4.html [Broken]
 
Last edited by a moderator:
  • #4
Don't give out solutions, g_edgar.
 
  • #5
D H said:
Don't give out solutions, g_edgar.

OK, deleted.
 
  • #6
DH

Thanks for your help. I am still a little confused where dy/dx comes into play, I will have to think about it. Maybe I will look in my 20yr old college text on differential equations to see if there is a similar probblem as g_edgar points out. I would guess the path is probably a parabola or hyperbola, maybe I can plug those into the equations.
 
  • #7
THe path is called a tractrix, or pursuit curve.

Where dy/dx comes into play is that it is much easier to derive the curve y=f(x) rather than to derive x and y as functions of time. From that, you can easily compute the point at which this curve intersects the x axis. Finally, given the chicken's speed, you can easily compute the time it takes for the fox to catch the chicken.
 
  • #8
D_H,

After thinking it over I think I understand now. The derivative dy/dx is the direction the fox is traveling. Therefore the solution would be solving this differential equation [tex] \frac {dy}{dx} = \frac {-y}{3t-x} [/tex] which I havn'e found the solution but that is OK. I just have bugged on how to solve it and I made it way more difficult than need be.

I did look in my college diff eq book and embarassing enough a similar problem is listed on page 12 (out of like 900+ total pages).
 
  • #9
For what its worth (not that it will assuage any pure mathematicians on this board), the problem was solved for all practical purposes when you made those two coupled ODE's.

They could be solved simultaneously by a variety of numerical methods.
 

1. What are differential equations of motion?

Differential equations of motion are mathematical equations that describe the relationship between the position, velocity, and acceleration of an object over time. They are used to model and predict the behavior of systems in motion, such as objects moving under the influence of forces.

2. Why are some differential equations of motion considered difficult?

Some differential equations of motion are considered difficult because they cannot be solved analytically and require more complex methods, such as numerical techniques, to find a solution. These equations may also involve multiple variables and parameters, making them more challenging to solve.

3. How are differential equations of motion used in science?

Differential equations of motion are used in science to model and understand the behavior of physical systems, such as the motion of planets, the trajectory of projectiles, and the vibrations of a pendulum. They are also used in engineering and technology to design and optimize systems, such as aircraft and automobiles.

4. What are some common techniques for solving difficult differential equations of motion?

Some common techniques for solving difficult differential equations of motion include separation of variables, integration by parts, and using numerical methods such as Euler's method or Runge-Kutta methods. Other techniques, such as Laplace transforms and Fourier series, can also be useful in certain situations.

5. Can differential equations of motion be solved without advanced mathematics?

While some differential equations of motion may require advanced mathematical techniques to solve, there are many simpler equations that can be solved using basic calculus and algebra. Additionally, there are numerous computer programs and tools available that can help solve difficult differential equations of motion without extensive mathematical knowledge.

Similar threads

  • Calculus
Replies
2
Views
1K
Replies
3
Views
1K
  • Introductory Physics Homework Help
Replies
13
Views
670
Replies
12
Views
2K
Replies
19
Views
2K
Replies
2
Views
695
  • Calculus
Replies
20
Views
3K
Replies
45
Views
2K
Replies
3
Views
1K
  • Introductory Physics Homework Help
Replies
17
Views
1K
Back
Top