Please help me with this problems, I would like to learn how to do them in clear

Question

Please help me with this problems, I would like to learn how to do them in clear steps if possible.

Q: Let a paratrooper of mass m jump from rest from a hovering helicopter at a height sufficient for the jumper to reach the terminal velocity vT and high enough for he air resistance to to be modeled as a quadratic drag:

Fair = - v2

The acceleeration due to gravity is approximated as g ~ 9.8 m/s2 ~ 10./s2. A simple dimensional analysis would lead to the equation of the form:

v(t) = vT * tan h (squrt[ g/m] t )

Try ti prove it using Newton's second law as the rate equation for v:

F= ma = mg - v2

where a = acceleration

a = v' = dv/dt

Note, VT = squrt[mg/] as it is the equilibrium of zero net acceleration a = dv/dt approached by v as the time moves forward. You may either 'solve in paramemter ' or you may use the following values:

m=60kg, =24 N*s2 /m2 = 24 kg/m. That would lead to v in m/s.

Hint: Can separate the varaible v and t :

integral dv / g- (/m) v2 = integral dt (the constant of intergration v(0)=0 )

Thank you for your time and effort in advance.

Explanation / Answer

               Given differential   equation   is

            F = ma = mg - v2

                         mdv/dt =   mg - v2

                      dv/dt =   g - v2/m

                 Now, On seperating   the variable ,We get

             dv/dt = g - v2/m

                  dv / [ g - v2/m ] = dt

               Now, integrating    this   equation wrt   v , We get

             Integral   dv / [ g - (/m)v^2 ] = Integral of   dt

                 arctanh [ sqrt (/m) v / sqrtg ] / sqrt [g (/m)] = t + C

                Now , initial conditions are    given v (0) = 0

              At    t = 0 ,   v = 0

               On Putting these values we get value of    C

    arctanh [ sqrt (/m) v / sqrtg ] / sqrt [g (/m)] = t + 0

              Finally we get

   arctanh [ sqrt (/m) v / sqrtg ] / sqrt [g (/m)] = t   

               arctanh [ sqrt (/m) v / sqrtg ] = sqrt [g (/m)] t

                             V(t) = tanh [sqrt [g (/m)]t] * [ sqrt { mg / ) ]

           Now, on Putting the value of m = 60kg, =24 N*s2 /m2 = 24 kg/m.

   We finally get

           V(t) = tanh [sqrt [10 ( 24/60)]t] * [ sqrt { 60*10 / 24 ) ]     

                             V(t) = tanh [sqrt [(4)]t] * [ sqrt { 600 / 24 ) ]  

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