Advanced Quadratic drag - How to solve?
So I have gotten this far...
#a=((F_a-(c_d*rho_h*A*v^2)/2)-m_t*g_r)#
This is based upon #F_a-(F_d+F_g)/m =a#
But #a# is #(dv)/(dt)#
So this becomes a differential equation because the faster you go the harder it is to go faster.
How can one solve this?
Please note I'm currently a Calc AB student.
So I have gotten this far...
This is based upon
But
So this becomes a differential equation because the faster you go the harder it is to go faster.
How can one solve this?
Please note I'm currently a Calc AB student.
1 Answer
Supposing that
we have
Explanation:
This equation
with
can be reduced to
which is the known Riccati equation.
https://en.wikipedia.org/wiki/Riccati_equation
Now making
a second order ODE which is
A solution
As an example, supposing that
we obtain
after solving
with general solution
Explanation . substituting in
we get
grouping
or
but
and
Finally putting all together
with