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A trace between a dog and a rabbit

  1. Oct 7, 2011 #1
    1. The problem statement, all variables and given/known data
    A dog is at a distance L due north of a rabbit. He starts to pursue the rabbit and its motion always points to the rabbit. Given that the rabbit keeps running due east with a constant speed v and the dog's speed is a constant u, where v<u. Find the time that the dog catches the rabbit according to the method stated below.

    (a) Consider the rabbit as a moving origin of a polar coordinates. Let [itex]\vec{r}[/itex] be the postition vector of dog relative to rabbit. Write down the velocity vectors relative to the rabbit along [itex]\vec{e_{r}}[/itex] [itex]\vec{e_{θ}}[/itex] respectively.

    (b) Show that r=[itex]\frac{L (cot \frac{θ}{2})^\frac{u}{v}}{sinθ}[/itex]

    (c) Use the result of (a) and (b) to find τ, the time for the dog to catch the rabbit.
    [Hint: Consider the relation τ=[itex]\int dt[/itex] and dt= [itex]\frac{dθ}{dθ'}[/itex]]

    2. Relevant equations
    u speed of the dog
    v speed of the rabbit
    θ angle measure from east to the position of the dog.
    L original dist. between the 2 animals


    3. The attempt at a solution
    For part (a), we can simply do it by resolving components.
    I get [itex]\vec{v}[/itex]=<u+vcosθ, vsinθ>

    But for part b and c, I have no idea.
    We need to integrate [itex]\vec{v}[/itex] with repest to t in order to get [itex]\vec{r}[/itex], but in part a, [itex]\vec{v}[/itex] depends on θ only.

    Alway, I don't know where can I use the hint.
     
  2. jcsd
  3. Oct 8, 2011 #2

    vela

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    Use the fact that [tex]\vec{v} = \dot{r}\vec{e}_r + r\dot{\theta}\,\vec{e}_\theta[/tex]and[tex]\frac{dr}{d\theta} = \frac{dr}{dt}\frac{dt}{d\theta}[/tex]
    Make sure you get the signs correct.
     
  4. Oct 8, 2011 #3
    Are you talking about part b?
    If yes, then what is [itex]\frac{dt}{dθ}[/itex]

    Is it [itex]\frac{1}{\dot{θ}}[/itex]?
    Then how to find [itex]\dot{θ}[/itex]?
     
  5. Oct 8, 2011 #4
    I finish part b now! :)
    But I don't know how to integrate the monster in part c.
     

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  6. Oct 8, 2011 #5

    vela

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    Isn't it supposed to be sin2 θ on the bottom?

    I'd try the substitution u = cot(θ/2) first.
     
    Last edited: Oct 8, 2011
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