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Constant velocity and vectors

  1. Jul 11, 2012 #1
    In this question, the horizontal unit vectors i and j are directed due east and north respectively.

    A coastguard station O monitors the movements of ships in a channel. At noon, the station's radar records two ships moving with constant speed. Ship A is at the point with position vector (−3i+10j)km relative to O and has velocity (2i+2j)kmh−1. Ship B is at the point with position vector (6i+j)km and has velocity (−i+5j)kmh−1.

    a Show that if the two ships maintain these velocities they will collide.

    The coastguard radios ship A and orders it to reduce its speed to move with velocity (i+j)kmh−1. Given that A obeys this order and maintains this new constant velocity.

    b find an expression for the vector AB→ at time t hours after noon,

    c find, to three significant figures, the distance between A and B at 1500 hours,

    d find the time at which B will be due north of A.

    a) done
    b) (9-2t)i + (-9+4t)j done
    c) done
    d) I'm not sure how to do this but I thought if B will be due north of A, then the i components should be the same no? so I done this i.e 10+t = 16 => t = 6 however this is wrong and in the solutions they say that the i component should be equal to 0, however in the next question:


    Two ships P and Q are moving along straight lines with constant velocities. Initially P is at a point O and the position vector of Q relative to O is (12i+6j)km, where i and j are unit vectors directed due east and due north respectively. Ship P is moving with velocity 6ikmh−1 and ship Q is moving with velocity (−3i+6j)kmh−1. At time t hours the position vectors of P and Q relative to O are p km and q km respectively.

    a Find p and q in terms of t.

    b Calculate the distance of Q from P when t=4.

    c Calculate the value of t when Q is due north of P.


    Part c) they say that it's when the i components are equal, which is what I done for the first question.

    Could anyone explain what is going on?
     
  2. jcsd
  3. Jul 11, 2012 #2

    Simon Bridge

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    (d) the vector [itex]\vec{AB}[/itex] will only have a [itex]\vec{j}[/itex] component.
     
  4. Mar 19, 2014 #3
    How did you get the answer for part b in that 1st question?
    I'm stuck on how to represent AB
     
  5. Mar 20, 2014 #4

    Simon Bridge

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    Welcome to PF;
    Vector addition.

    Represent the vectors by arrows on a bit of graph paper and work it out.
    AB is the vector pointing from point A to point B.
    How would you normally do that?
     
  6. Mar 20, 2014 #5
    I would use i+I notation. But I can't be sure that
     
  7. Mar 20, 2014 #6
    Sorry. Message accidentally sent before finished
     
  8. Mar 20, 2014 #7
    I can't be sure that the position of the ships haven't changed
     
  9. Mar 20, 2014 #8

    Simon Bridge

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    The positon of the ships does change - their relative position changes too.
    So you need to represent their positions as a vector that changes in time.

    i.e. if ship A travelled at speed 3kmph 45deg from the +x axis starting at position ##\vec a_0= 3\hat\jmath##km, then it's position vector will be ##\vec a = (3\hat\imath + (3+3t)\hat\jmath)##km.
     
  10. Mar 20, 2014 #9
    I'm not sure what hat means
     
  11. Mar 20, 2014 #10

    Simon Bridge

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    indicates a unit vector:
    ##\hat\imath = \mathbf{i}##, ##\hat\jmath = \mathbf{j}##, ##\hat k = \mathbf{k}##
    ... are the Cartesian unit vectors.

    In general: $$\hat v = \frac{\vec v}{v}$$
     
  12. Mar 20, 2014 #11
    The book doesn't mention hat. It just says that the coastguard radios ship A to reduce speed to i+j kmh-1
     
  13. Mar 20, 2014 #12

    Simon Bridge

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    Do you know what the i and j mean?
     
  14. Mar 20, 2014 #13
    It's fine. I got an answer. Thank you
     
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