A submarine of mass m is travelling at max power and has engines that deliver a force F at max power to the submarine. The water exerts a resistance force proportional to the square of the submarine's speed v.The submarine increases its speed from v 1 to v 2 , show that the distance travelled in this period is m 2 k ln ⁡ F − k v 1 2 F − k v 2 2 where k is a constant.I've found d v d t = 1 m ( F − k v 2 ) using ∑ F = m a but I am struggling to progress further using integration or the fact that d v d t = v d v d x = d d x ( 1 2 v 2 ) = d 2 x d t 2 which is how I've been taught to solve resisted motion questions.

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A submarine of mass m is travelling at max power and has engines that deliver a force F at max power to the submarine. The water exerts a resistance force proportional to the square of the submarine&#039;s speed v.The submarine increases its speed from       v    1   to       v    2  , show that the distance travelled in this period is       m          2      k        ln  ⁡            F      −      k              v        1        2                    F      −      k              v        2        2             where k is a constant.I&#039;ve found             d      v              d      t        =      1    m    (  F  −  k      v    2    ) using   ∑      F    =  m  a but I am struggling to progress further using integration or the fact that             d      v              d      t        =  v            d      v              d      x        =      d          d      x        (      1    2        v    2    )  =                    d        2            x              d              t        2             which is how I&#039;ve been taught to solve resisted motion questions.

charringpq49u · Accepted Answer

You are close:            m      v              F      −      k              v        2                        d      v              d      x        =      m          −      2      k                  d              (        F        −        k                  v          2                )                    F      −      k              v        2                  1          d      x        =  1And therefore the desired result by integration.

kwisangqaquqw3 · Accepted Answer

Using chain rule,                     d        x                    d        v              =                  d        x                    d        t              ⋅                  d        t                    d        v              =          v      a      So,   d  x  =                                                  v                                                                  a                      ⋅  d  v  =                                                  m          v                                                                  F          −          k                      v            2                                ⋅  d  vIntegrating both sides and given v increases from       v    1   to       v    2  ,      x    =    −          m              2        k                   ln    ⁡          (                        F          −          k                      v            2            2                                    F          −          k                      v            1            2                              )        =          m              2        k                   ln    ⁡          (                        F          −          k                      v            1            2                                    F          −          k                      v            2            2                              )

A submarine of mass m is travelling at max power and has engines that deliver a force F at max power

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