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River problem (calculus)

  1. Mar 31, 2009 #1
    1. The problem statement, all variables and given/known data

    find the optimal angle so that the time required for the swimmer,runner to cross the river (directly opposite starting position) is minimum. the swimmer swims to one point at the end of the river then he/she will run to the target(opposite from starting position)

    swim speed 4miles/hour
    running speed 10miles/hour
    river flow speed 2miles/hour

    this is all the information that is given

    2. The attempt at a solution

    http://img13.imageshack.us/img13/2500/extracredit.jpg [Broken]

    h=width of river
    k=distance for running part of problem
    z=distance covered by the resultant velocity
    [tex]\theta[/tex]= angle swimmer takes off
    Vswim=velocity swimmer swims
    Vrun=velocity he/she run
    Vriver=velocity of river flow
    Vres=resultant velocity of river flow and swimming

    ok so here is what i have so far,

    Vres2=[Vswim*sin(90-[tex]\theta[/tex])]2+[Vriver-Vswim*(cos (90-[tex]\theta[/tex])]2

    therefore:

    Vres2=[Vswim*cos[tex]\theta[/tex]]2+[Vriver-Vswim*sin[tex]\theta[/tex]]2


    it is also true from diagram that
    z2=h2+k2

    now:

    equation 1 and 2

    z2=[Vres*t]2=t2*Vres2

    k=Vrun*t


    ok....

    it is also true that

    equation 3:
    z2=h2+k2

    sub equation 1 and 2 into equation 3


    ... here is where i am stuck...

    what do i do now
    i can only think of differentiating d(theta)/d(t) and set that to zero... is that even correct i have no idea where to go on from this even when i try d(theta)/d(t) and i differentiate the equation i am still left with the varible time...


    this is an interesting problem. and i would like to know how to solve it

    any help would be appreciated , thank you
    vthenry
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Mar 31, 2009 #2

    LowlyPion

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    Welcome to PF.

    Since time is the variable that needs to be optimized, I'd suggest working in equations that represent the times of the segments involved. Then you can optimize that function in t, to find its minimum as you suggest by taking the derivative.

    For instance Time to swim across will be the distance h divided by his y component of velocity. Then figure his time to run the distance k along the bank as that time to cross times the river speed less whatever angle against the current he swam and all that divided by his running velocity on land. (If he swims at an angle with the current then this will necessarily be longer so you can discard that as an option if you want his minimum time.)
     
  4. Mar 31, 2009 #3
    hey thanks for helping =)

    ill try it and let you know how i do

    cheers,
    vthenry
     
  5. Mar 31, 2009 #4
    hey i dont understand this part
    "as that time to cross times the river speed less whatever angle against the current he swam and all that divided by his running velocity on land"

    thanks
     
  6. Mar 31, 2009 #5

    LowlyPion

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    The distance k is determined by the speed the swimmer is carried down the river. The first term. The second term takes account of the angular component of his velocity that is || to the direction of the current. That may be either a + or -, but since you are wanting to minimize the time you would take (-) that component times the time right?

    The distance k/Vrunning is the time it takes to run along the other bank.
     
  7. Mar 31, 2009 #6
    ok so, i have this equation now which i have to take the derivative of

    T_total = k/Vrun + h/(Vswim(y component))

    k=T_swim(Vriver-Vswim(x component))
     
  8. Mar 31, 2009 #7

    LowlyPion

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    That looks about right.

    Now you should have T(θ), so you need to figure the minimum for that function.
     
  9. Mar 31, 2009 #8
    wow the equation i have to solve is too difficult
    set dt/d(theta) = 0 and i have this eqaution T(theta)

    T=t_swim+t_run

    T=[tex]\frac{h}{Vswim*cos\theta}[/tex] + ([tex]\frac{Vriver-Vswim*sin\theta}{Vrun}[/tex])*tswim

    how am i going to solve this now for the optimal angle to minimize time ?
     
    Last edited: Mar 31, 2009
  10. Mar 31, 2009 #9

    LowlyPion

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    It looks to me that you can differentiate this with respect to θ and set it to 0 to determine the minimum of the function T(θ)

    Sub in the values given in the problem and rewrite the terms in terms of Sec and Tangent and then differentiate right?
     
  11. Mar 31, 2009 #10

    LowlyPion

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    Your equation should likely be simplified further by the way to

    T = h/(Vsm*cosθ ) * (1 + (Vrvr - Vsw*sinθ)/Vrun )
     
  12. Mar 31, 2009 #11

    ok somehow after differentiating and setting d(t)/d(theta) = 0

    and some algebraic minipulation i am left with this

    cos(theta)+sin(theta)tan(theta)+tan(theta)=0

    i have no idea if this is correct or not.. but how do i solve this now ?
     
  13. Mar 31, 2009 #12

    LowlyPion

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    I converted my equation to

    T = h/4*(secθ) + 2h/40*secθ - 4*h/40*tanθ

    That yields

    T = 12*h*secθ/40 - h*tanθ /10

    Now just differentiate that and set it to 0.
     
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