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Relative Velocity collsion

  1. Feb 22, 2013 #1
    This isn't really homework its an example, But I need help understanding it to do my homework

    1. The problem statement, all variables and given/known data
    A destroyer is 500 km due west of a frigate. The destroyer is travelling at 10 km/h in a direction of 30 degrees north of east.The frigate is travelling at 5(2)^(1/2) in a NW direction.

    (i) Find the velocity of the frigate relative to the destroyer.

    (ii) Show that they are on a collision course.

    (iii) When will they collide?

    2. Relevant equations
    Vab = Va -Vb (V is a vector I don't know how to write vector notation on a computer)
    d is the destroyer
    f is the frigate
    3. The attempt at a solution
    (solution of example as in the book)
    (i)Vd = 10cos30i + 10sin30j
    = 8.66i + 5j

    Vf = -5(2)^(1/2)cos45i + 5(2)^(1/2)sin45j = -5i +5j

    Vfd = (-5i +5j) - (8.66i +5j)
    =-13.66i

    (ii)
    Position of frigate relative to the destroyer is at 500i km
    we write this as Rfd = 500i km

    the velocity of the frigate relative to the destroyer is
    Vfr = -13.66i km/h (I think this is an error and it should Vfd)

    since Vfd = -k(Rfd) where k is a positive constant , they must be on a collision course.

    (iii)
    the time of the collision is given by relative distance/ relative speed
    500/13.66 = 36 hours and 36 minutes later.


    I understand (i) completely my main problem lies with (ii) where it says
    "since Vfd = -k(Rfd) where k is a positive constant , they must be on a collision course."
    The only way I can see they could make them both equal is if k were the inverse of time and if it is then why not just put k on the left and let it equal to time. I also don't get why the negative sign is needed.If anybody could help explain this one line to me it would be greatly appreciated.
     
    Last edited: Feb 22, 2013
  2. jcsd
  3. Feb 22, 2013 #2

    haruspex

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    It doesn't matter whether you put k on the right and have it as an inverse of time, or put it on the left and have it represent (the more obvious) collision time. The point is merely that (with the minus sign) the ratio is positive: i.e. the distance is diminishing in magnitude, so the time at which they will be at the same point is in the future.
     
  4. Feb 22, 2013 #3
    Thanks I get it now its just that generally when I see constants of proportionality there usually put on the side of the equation where they wouldn't be inverted.
     
  5. Feb 22, 2013 #4
    Sorry to bother you again but it would be fine if I said -k(Vfd) = Rfd wouldn't it?
    since k would now mean time and the ratio between them would still be positive.
     
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