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Vectors with a dose of arithmetic :3

  1. May 19, 2014 #1
    1. The problem statement, all variables and given/known data
    Two velocities acting at a particular point are such that:
    • The sum of their respective magnitudes is 15m/s
    • The product of their respective magnitudes are 56m2/s2
    • The resultant is 13m/s.
      Find the two velocities and the angle between them.

    2. Relevant equations
    [itex]R^2 = P^2 + Q^2 + 2PQcos\Theta [/itex]

    3. The attempt at a solution

    At first I tried to change the statements into equations:
    [itex]P + Q = 15m/s[/itex]
    [itex]P * Q = 56m^2/s^2[/itex]
    [itex]R = 13m/s[/itex]

    And then I used the binomial expressions knowledge to try solving it:
    [itex](P+Q)^2 = P^2 + 2PQ+Q^2[/itex]
    [itex]15^2 = P^2 + 2*56 + Q^2[/itex]
    [itex]225 = P^2 + Q^2 + 112 [/itex]
    [itex]P^2 + Q^2 = 225 - 112[/itex]
    [itex]P^2 + Q^2 = 113[/itex]

    I then substituted the value obtained above for the following equation:
    [itex]R^2 = P^2 + Q^2 + 2PQcos\Theta [/itex]
    [itex]13^2 = 113 + 2 X 56 X cos\Theta[/itex]
    [itex]169 - 113 = 112cos\Theta[/itex]
    [itex]56 = 112cos\Theta[/itex]
    [itex]cos\Theta = 56/112[/itex]
    [itex]cos\Theta = 1/2[/itex]
    [itex]\Theta = 60°[/itex]

    Through this I could find the value of the angle between them. And also, that [itex]P^2 + Q^2 = 113[/itex]. But I can't seem to find the separate values for P and Q. Any ideas? Thank you :3
     
    Last edited: May 19, 2014
  2. jcsd
  3. May 19, 2014 #2
    USE THE EQUATION :-
    (P-Q)2 = (P+Q)2 - 4PQ
    To find the difference between their magnitudes, after then you can simply use elimination method to solve the two equations you got.
     
  4. May 19, 2014 #3
    (P+Q) P= P2 + PQ

    15 P = P2 + 56

    Solve that quadratic equation.
     
  5. May 19, 2014 #4
    Oh thank you!
    I got the final result as P = 7/8, Q = 7/8. Thank you :3
     
  6. May 19, 2014 #5

    SteamKing

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    Science Advisor
    Homework Helper

    I think you mean P = (7, 8) m/s and Q = (7, 8) m/s

    Writing P and Q the way you did makes it look like a fraction, which is confusing.
     
  7. May 19, 2014 #6
    Don't you mean either P=7 and Q=8 or P=8 and Q=7 ?
     
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