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Vector expressions - equivalence

  1. Jan 3, 2009 #1
    1. The problem statement, all variables and given/known data
    A car is travelling at 12ms^-1. To the passenger in the car the wind appears to be blowing at 8.0ms^-1 at right angles to the road. What is the magnitude and direction of the velocity of the wind with respect to the ground.


    2. Relevant equations
    I can think of two possible vector expressions which should be equivalent. I am clearly doing something wrong because they are not.
    Using:
    vwg = velocity of wind wrt ground
    vcg = velocity of car wrt ground
    vwc = velocity of wind wrt car
    Expression 1
    vwg = vwc + vcg

    Expression 2 (following the rule of subtracting the observer's movement)
    vwc= vwg-vcg

    Yes - I'm aware I should add the negative vector and that the negative sign means the reverse direction

    3. The attempt at a solution

    Expression 1:
    The solution vector diagram is (excuse the dots so I don't lose the spaces):

    ^----------->vcg
    |..............^
    |vwc ..... /
    |........ / vwg
    |....../
    |../
    |/
    It gives a vwg magnitude of c. 14ms^-1 and a direction of tan^-1(8/12) = c. 34 degrees with the wind coming from behind the car

    Expression 2:
    The solution vector diagram is:

    <----------------- -vcg
    .^.......................^
    ....\....................|
    ........\ ...............| vwc
    ....vwg..\.............|
    ..............\..........|
    ..................\......|
    ......................\..|
    .........................\|
    Of course the magnitude is the same as in the previous example but the direction is
    tan^-1 (8/-12) = -34 degrees.

    My concern is that the wind appears to be coming from the front!

    Gloom - what have I misunderstood?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jan 4, 2009 #2
    Your method is correct. The answer is not exactly 14m/s. In both cases it is only the direction you start with that results in the -ive or +ive sign. Why do you think the wind appears to be coming from the front?
     
  4. Jan 4, 2009 #3
    Thank you for your reassurance. I lacked the confidence to be certain the second vector diagram was the equivalent of the first - especially because vector arithmetic is new to me. I would like to ask this naive question - if a, b and c are vector quantities and they are related:
    a = b+c
    then is it correct to transpose so that:
    b= a-c
    ?

    Why did I think the wind was coming from the front? The honest answer is twofold: first: sometimes I have a pathological inability to imagine the behaviour of the real word and, second I had convinced myself the second expression/vector diagram was telling a different story to the first.
     
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