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This is just to share some teaching experience.
A problem about trajectory of a projectile in the atmosphere is very well known. The trajectory has a vertical asymptote. This fact is easy to obtain by using the stability theory.
Consider equations of motion. Let ##\boldsymbol v=v_x\boldsymbol e_x+v_y\boldsymbol e_y## be the velocity of the projectile relative the standard Earth fixed frame ##xy## with ##y## directed upwards.
The second Newton
##m\boldsymbol {\dot v}=m\boldsymbol g-\gamma |\boldsymbol v|\boldsymbol v;##
implies
##m\dot v_x=-\gamma v_x\sqrt{v_x^2+v_y^2},\quad \gamma=const>0,##
##m\dot v_y=-mg-\gamma v_y \sqrt{v_x^2+v_y^2}.##
This system has an equilibrium: ##v_x=0,\quad v_y=v^*:=-\sqrt{mg/\gamma}.##
This equilibrium is exponentially stable: eigenvalues of linearized system around this equilibrium are both negative so that
##v_x=O(e^{-c_x t}),\quad v_y=v^*+O(e^{-c_yt}),\quad c_x,c_y>0,\quad t\to\infty;##
and
##x(t)=x_0+\int_0^tv_xd\tau\to const,\quad y(t)\sim v^*t.##
Besides the quadratic model of atmosphere drag there are a lot of other models where the stability theory works but an explicit integration is impossible
A problem about trajectory of a projectile in the atmosphere is very well known. The trajectory has a vertical asymptote. This fact is easy to obtain by using the stability theory.
Consider equations of motion. Let ##\boldsymbol v=v_x\boldsymbol e_x+v_y\boldsymbol e_y## be the velocity of the projectile relative the standard Earth fixed frame ##xy## with ##y## directed upwards.
The second Newton
##m\boldsymbol {\dot v}=m\boldsymbol g-\gamma |\boldsymbol v|\boldsymbol v;##
implies
##m\dot v_x=-\gamma v_x\sqrt{v_x^2+v_y^2},\quad \gamma=const>0,##
##m\dot v_y=-mg-\gamma v_y \sqrt{v_x^2+v_y^2}.##
This system has an equilibrium: ##v_x=0,\quad v_y=v^*:=-\sqrt{mg/\gamma}.##
This equilibrium is exponentially stable: eigenvalues of linearized system around this equilibrium are both negative so that
##v_x=O(e^{-c_x t}),\quad v_y=v^*+O(e^{-c_yt}),\quad c_x,c_y>0,\quad t\to\infty;##
and
##x(t)=x_0+\int_0^tv_xd\tau\to const,\quad y(t)\sim v^*t.##
Besides the quadratic model of atmosphere drag there are a lot of other models where the stability theory works but an explicit integration is impossible
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