Results 1 to 7 of 7

Math Help - Differential equation--Moth problem

  1. #1
    Junior Member
    Joined
    Jan 2010
    Posts
    52

    Differential equation--Moth problem

    One theory about the behaviour of moths states that they navigate at night by keeping fixed angle between their velocity vector and the direction of the Moon [or some bright star]. A certain moth flies near to a candle and mistakes it for the Moon. What will happen to the moth?
    Hints: in polar coordinates (r,θ ), the formula for the angle ω between the radius vector and the velocity vector is given by



    Use the formula to solve for r as a function of .

    I do not know how to set up the necessary equations. I tried sketching a diagram,but I cannot seem to find a relation between ω and θ necessary to solve the formula given. Is my diagram correct? Thank you for your help.

    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4

    Re: Differential equation--Moth problem

    Quote Originally Posted by cyt91 View Post
    One theory about the behaviour of moths states that they navigate at night by keeping fixed angle between their velocity vector and the direction of the Moon [or some bright star]. A certain moth flies near to a candle and mistakes it for the Moon. What will happen to the moth?
    Hints: in polar coordinates (r,θ ), the formula for the angle ω between the radius vector and the velocity vector is given by



    Use the formula to solve for r as a function of .

    I do not know how to set up the necessary equations. I tried sketching a diagram,but I cannot seem to find a relation between ω and θ necessary to solve the formula given. Is my diagram correct? Thank you for your help.

    First take the position of the candle as the origin.

    CB
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5

    Re: Differential equation--Moth problem

    What CB suggested is excellent!... to proceeding we suppose that the speed of the moth is in modulus a constant v, so that only its direction can be changed, and we indicate with \alpha the 'fixed angle' defined in the original post. Setting the position of the moth as a complex number...

    z(t)= x(t)+ i\ y(t) = r(t)\ e^{i \theta(t)} (1)

    ... the complex equation describing the flight of the moth is...

    z^{'}= (r^{'} + i\ r\ \theta^{'})\ e^{i\ \theta}= v\ e^{i\ (\theta- \frac{\pi}{2} + \alpha)} \rightarrow -r\ \theta^{'} +i\ r^{'} = v\ e^{i\ \alpha} (2)

    The (2) is equivalent to a system of two DE in two variables...

    r^{2}\ \theta^{'\ 2} + r^{'\ 2}= v^{2}

    \frac{r^{'}}{r\ \theta^{'}}= - \tan \alpha (3)

    ... and a sucessive post is dedicated to investigate about the solution of (3)...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5

    Re: Differential equation--Moth problem

    For semplicity sake we set v=1 so that the system of DE becomes...

    r\ \theta^{'}=- \cos \alpha

    r^{'}=\sin \alpha (1)

    Of course the solution second DE is immediate...

    r(t)= t\ \sin \alpha + c_{1} (2)

    ... and that means that...

    a) for 0< \alpha< \frac{\pi}{2} the distance of the moth from the candle will increase without limits...

    b) for \alpha=0 the distance of the moth from the candle remains constant and the motion is 'uniform circular'...

    c) for -\frac {\pi}{2}<\alpha<0 the distance of the moth from the candle vanishes at the time - \frac{c_{1}}{\sin \alpha} and the moth is 'kaput'...

    Now if we substitute (2) in the first DE we obtain...

    \theta^{'}= - \frac{\cos \alpha}{t\ \sin \alpha + c_{1}} (3)

    .... and the solution is...

    \theta(t)= \frac{1}{\tan \alpha}\ \ln \frac{1}{t\ \sin \alpha\ + c_{1}} + c_{2} (4)

    Kind regards

    \chi \sigma
    Last edited by chisigma; January 15th 2012 at 08:44 AM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4

    Re: Differential equation--Moth problem

    Being in the trade I know that we do not normally go for an explicit solution, but observe that the range decreases at a constant rate, and that the angle rate goes to infinity as range goes to zero (as is obvious from chisigma's equations). Also since such systems are usually (body) rate limited you can calculate the range at which the lead-pursuit model breaks down (when you should end up circling the flame).

    CB
    Last edited by CaptainBlack; January 15th 2012 at 10:17 AM.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5

    Re: Differential equation--Moth problem

    Quote Originally Posted by CaptainBlack View Post
    Being in the trade I know that we do not normally go for an explicit solution, but observe that the range decreases at a constant rate, and that the angle rate goes to infinity as range goes to zero (as is obvious from chisigma's equations). Also since such systems are usually (body) rate limited you can calculate the range at which the lead-pursuit model breaks down (when you should end up circling the flame).

    CB
    I suppose You intend the case c) , when is - \frac{\pi}{2}<\alpha<0. If the hypothesis of speed constant in modulus is true, then in such a situation when the decreasing of the distance of the moth from the flame produces a great increasing of the angular speed so that it is not surprising that...

    \lim_{t \rightarrow t_{0}} \theta (t)= - \infty\ ,\ t_{0}= - \frac{c_{1}}{\sin \alpha}

    Of course different hypothesis about the speed produce a different equation and a different scenario...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4

    Re: Differential equation--Moth problem

    Quote Originally Posted by chisigma View Post
    I suppose You intend the case c) , when is - \frac{\pi}{2}<\alpha<0. If the hypothesis of speed constant in modulus is true, then in such a situation when the decreasing of the distance of the moth from the flame produces a great increasing of the angular speed so that it is not surprising that...

    \lim_{t \rightarrow t_{0}} \theta (t)= - \infty\ ,\ t_{0}= - \frac{c_{1}}{\sin \alpha}

    Of course different hypothesis about the speed produce a different equation and a different scenario...

    Kind regards

    \chi \sigma
    I think either you are using a different convention from me about which angle is \alpha or you have the trig functions reversed. I think \dot{r}=-\cos(\alpha), where \alpha is the angle from the moth's velocity vector to the line-of-sight to the candle (when \alpha is zero the moth flies directly towards the candle).

    The cases with -\frac{\pi}{2}<\alpha<\frac{\pi}{2} are all closing, and changing the sign of \alpha gives the mirror image trajectory. The remaining cases are all opening or constant range trajectories.

    Now because we are talking about moths I have assumed we are only interested in closing trajectories, as otherwise we would not be aware of the behavior.

    CB
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: January 17th 2012, 08:13 AM
  2. Replies: 1
    Last Post: April 11th 2011, 01:17 AM
  3. Differential equation problem
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: March 22nd 2009, 02:25 AM
  4. Differential Equation Problem
    Posted in the Differential Equations Forum
    Replies: 5
    Last Post: February 12th 2009, 05:12 PM
  5. Problem with a differential equation
    Posted in the Calculus Forum
    Replies: 7
    Last Post: April 29th 2008, 01:11 PM

Search Tags


/mathhelpforum @mathhelpforum