Speed of a parametric function?I know speed = |velocity|Why is speed of parametric defined as s p e e d = ( d x d t ) 2 + ( d y d t ) 2 How is this derived? What is the principle here? Is this Pythag? Thinking of this in terms of vectors?

Question

Speed of a parametric function?I know speed = |velocity|Why is speed of parametric defined as  s  p  e  e  d  =                    (                              d            x                                d            t                          )            2        +                  (                              d            y                                d            t                          )            2      How is this derived? What is the principle here? Is this Pythag? Thinking of this in terms of vectors?

vrtuljakwb · Accepted Answer

Step 1When you have a parametric equation in 2D it is usually defined as   α  (  t  )  =  (      α          1        (  t  )  ;      α          2        (  t  )  ).The velocity vector at time       t          0       would then be the derivative of   α at       t          0      , which is,      α    ′    (      t          0        )  =      lim          t      →              t                  0                                        α        (        t        )        −        α        (                  t                      0                          )                    t        −                  t                      0                               Assuming that       α          1        (  t  ) and       α          2        (  t  ) are both derivable, you get      α    ′    (      t          0        )  =      lim          t      →              t                  0                                (                                    α                      1                          (        t        )        −                  α                      1                          (                  t                      0                          )                    t        −                  t                      0                                ;                              α                      2                          (        t        )        −                  α                      2                          (                  t                      0                          )                    t        −                  t                      0                                          )      Step 2Using the fact that       lim          x      →              x                  0                                (        a  (  x  )  ;  b  (  x  )            )        =            (            lim          x      →              x                  0                      a  (  x  )  ;      lim          x      →              x                  0                      b  (  x  )            )      , you have       α    ′    (      t          0        )  =            (            α          1        ′    (      t          0        )  ;      α          2        ′    (      t          0        )            )      What you call speed at time       t          0       is in fact the norm of the velocity vector,       α    ′    (      t          0        ). And so,  speed  ⁡  (      t          0        )  =      |        |        α    ′    (      t          0        )      |        |    =      (          α              1            ′        (          t              0              )          )              2              +    (          α              2            ′        (          t              0              )          )              2

Speed of a parametric function? I know speed = |velocity| Why is speed of parametric defined as

Answered question

Answer & Explanation

New Questions in Integral Calculus