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Velocity and acceleration

  1. Oct 14, 2007 #1
    1. The problem statement, all variables and given/known data

    The position of a particle at time t is given by
    r(t) =<t,t^2,(2/3)t^3>

    (a) Find the velocity v, speed |v| and acceleration a of the particle, and the
    curvature k of its path, as a function of t.


    (b) Find the arc length function s(t), and the arc length of the curve as t increases
    from t = 0 to t = 3.

    (c) At the point <1,1,2/3>, find the unit tangent, principal normal and binormal
    vectors T, N and B, and find the curvature k and radius


    2. Relevant equations
    *NOTE* | | arround somethign mean the magnitude of the vector not absolute value
    IE |2i,3j,4k| = sqrt(4+9+16)
    Code (Text):

    T(t)= r'(t)
           |r'(t)|

    k=|r'(t)Xr"(t)|
           |r'(t)|^3

    N(t)=T'(t)
          |T'(t)|

    T(t)=r'(t)
         |r'(t)|

    B(t)=T(t)XN(t)

    r'(t)=v
    r"(t)=v'(t)=a
     
    3. The attempt at a solution



    a) r(t) =<t,t^2,(2/3)t^3 (given)
    v= r'(t)=1 i, 2t j, 2t^2 k
    a= r"(t)=v'(t)=0 i, 2 j, 4t k


    |v|=|r'(t)|=sqrt(1+4t^2+4t^4)=(2t^2+1)=|V|
    Code (Text):

    k=|r'(t)xr"(t)|
         |r'(t)|^3

    r'(t) X r"(t)= | i    j    k  |
                 |1  2t   2t^2  |  =[B](4t^2)i, -4t j, 2 k[/B]
                 |0   2   4t    |

    |r'(t) X r"(t)|= sqrt(16t^4+16t^2+4)
    =2sqrt(4t^4+4t^2+1)
    =2sqrt(2t^2+1)^2
    = [B]2(2t^2+1)[/B]

    k= 2(2t^2+1)
       (2t^2+1)^3

    [B]= 2
    (2t^2+1)^2[/B]
     

    did i do this right?

    s(t)=(integral form a to b of) |r'(t)|
    s(t)=(int of)2t^2+1

    integral form 0 to 3 is (2/3)t^3+t = (2/3)3^3 +3 = 18+3=21???


    c)At the point <1,1,2/3>, find the unit tangent, principal normal and binormal
    vectors T, N and B, and find the curvature k and radius


    my attempt....
    Code (Text):

    T(t)=r'(t)             r'(t)=1 i, 2t j 2t^2 k
         |r'(t)|             |r'(t)|=2t^2+1

    T(t)=(1/(2t^2+1)) i + ((2t)/(2t^2+1)) j + ((t^2)/(2t^2+1)) k

    the point 1,1,2/3 means t=1
    [B]T(1)=1/3 i + 2/3 j + 1/3 k[/B]  
     
    CORRECT!?!?!



    IM STUCK i can't find T'(t)

    but i found |T'(t)| with
    Code (Text):

    |T'(t)|=|r'(t) X r"(t)|      =       2(2t^2+1)          =   2
                 |r'(t)|^2              (2t^2+1)            2t^2+1
     
    also i found k with the solution from part a


    k= 2
    (2t^2+1)^2


    k=2/9????

    and once i get that T'(t) i can easy use B(t)=T(t) X N(t) to finish the problem

    so please point out any errors and help me figure out how to get T'(t) :D
     
    Last edited: Oct 14, 2007
  2. jcsd
  3. Oct 14, 2007 #2
    Your v x a is incorrect
     
  4. Oct 14, 2007 #3
    look better now?
     
    Last edited: Oct 14, 2007
  5. Oct 14, 2007 #4
    ok i fixed it, does everythign else look k correct? (b also)? 'cuse im running onto problems on C
     
  6. Oct 14, 2007 #5
    dont mean to be pushy but i need to sleep soon, can any1 shed some light please?
     
  7. Oct 15, 2007 #6
    bed in 20 minutes...any1?
     
  8. Oct 15, 2007 #7
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