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Integrating Velocity When in Unit Vector Notation

  1. Feb 5, 2014 #1
    1. The problem statement, all variables and given/known data

    Say for example, a particles velocity was given by the following equation:

    [itex]\vec{V}[/itex](t) = (2t2-4t3)[itex]\hat{i}[/itex] - (6t +3)[itex]\hat{j}[/itex] + 6[itex]\hat{k}[/itex]

    If I wanted to find the displacement of the particle between t=1s and t=3s, could I just integrate like this?

    [itex]\int \vec{V}[/itex]= (2t3/3 - t^4)[itex]\hat{i}[/itex] - (3t2 +3t)[itex]\hat{j}[/itex] + 6t [itex]\hat{k}[/itex] evaluated between 1.00 and 3.00

    = (-63i)-36j + 18k)-(2/3-1)i+(6j)-6k= -63.3i - 30j + 12k.

    Is this the correct way to do this?


    1. The problem statement, all variables and given/known data

    N/A


    2. Relevant equations

    N/A
     
    Last edited: Feb 5, 2014
  2. jcsd
  3. Feb 5, 2014 #2
    Yep, that's correct.

    As for why it's correct, suppose the particle's velocity was just 6i, so the distance is only changing in the i direction so you only integrate in that direction. Then if it's velocity was 6i + 3j, the total displacement is the same as moving the i component, then travelling in the j component separately.

    The total displacement is just the vector sum, hence why your integration is correct.
     
  4. Feb 5, 2014 #3
    So can you just treat each unit vector separately and integrate and evaluate each individually, then combine them all to find the displacement vector?

    Thanks!
     
  5. Feb 5, 2014 #4
    Yes, you can.
     
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