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Difficult System of Equations

  1. Apr 1, 2013 #1
    Hi all,

    I'm stuck on a system of equations I am left with at the end of a dynamics problem.

    [itex]
    a=-b+v
    [/itex]
    [itex]
    x=-b+vcos(30)
    [/itex]
    [itex]
    y=vsin(30)
    [/itex]
    [itex]
    171.5=20b^2+5a^2
    [/itex]
    [itex]
    a^2=x^2+y^2
    [/itex]
    [itex]
    10x-40b=0
    [/itex]
    All in degrees.
    I know I have an extra equation, but I thought I'd include it in case it is easier to solve with a certain five of them. I know it's not as simple as solving for one variable in terms of one other, plugging in and solving. I tried looking at it for ways to recombine the equations- no luck.

    Thank you,
    Jack
     
  2. jcsd
  3. Apr 1, 2013 #2

    berkeman

    User Avatar

    Staff: Mentor

    Could you clarify which quantities are variables, and which are constants? Is v a variable, for instance?
     
  4. Apr 2, 2013 #3

    Ray Vickson

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    Science Advisor
    Homework Helper

    You have 6 equations in 5 unknowns a, b, v, x, y, In this case, there is no solution: the equations are inconsistent.

    If you leave out the fifth equation (##a^2 = x^2 + y^2##) you can fairly easily solve the remaining equations, just by expressing a,v,x,y in terms of b (using equations 1,2,3,6) and then finding b from equation 4. To see that the original system is inconsistent, just substitute the resulting solution into the missing equation 5.
     
  5. Apr 5, 2013 #4
    I didn't really go into details to calculate the answer, and with 6 equations to find 5 variables it is more than enough and I assume that all equations are correct (i.e. either equation left out can provides same answer), let's use a coordinate system to solve this problem, plot x-y axis and for equation 5, a^2=x^2+y^2, it is a circle, from there you can easily get your answer.

    Let's explain more, equation 2, x-b=vcos30, equation 3 y=vsin30, combine can get (x-b)^2+y^2=v^2, another circle, I think this can gives you a clear picture?
     
    Last edited: Apr 5, 2013
  6. Apr 5, 2013 #5

    Ray Vickson

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    Science Advisor
    Homework Helper

    Perhaps you did not read my reply: the equations are inconsistent, so they cannot all be correct.
     
  7. Apr 7, 2013 #6
    I think you should really read what I wrote, equation 2 and 3 actually can combine to gives (x+b)^2+y^2=v^2. So there are 5 equations, 5 variables. I don't know where your inconsistent come from, and why they cannot be all correct.

    Okay.. yeah previous one I wrote wrongly, it should be x+b instead of x-b.
     
  8. Apr 7, 2013 #7
    a=−b+v (1)
    x=−b+vcos30 (2)
    y=vsin30 (3)
    171.5=20b^2+5a^2 (4)
    a^2=x^2+y^2 (5)
    10x−40b=0 (6)

    From equation 6 => x=4b (7)
    From equation 2 => x+b=vcos30 (8)
    square(equation 8) + square(equation 3) => (x+b)^2+y^2=v^2 (9)

    Sub equation 7 to equation 9 => 25b^2+y^2=v^2 (10)
    Sub equation 7 to equation 5 => 16b^2+y^2=a^2 (11)
    equation 10 minus equation 11 => 9b^2=v^2-a^2 (12)

    From equation 1 => v=a+b (13)
    Sub equation 13 to equation 12 => 9b^2=(a+b)^2-a^2
    8b^2-2ab=0
    b=0 or a=4b

    Okay I think Ray is right, my view of the "combining 2 equations" become a circle is a big mistake because of the angle of 30 degree, they are 2 linear equations and a combination of them can't make a circle.

    Sorry for the stupid concept.
     
    Last edited: Apr 7, 2013
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