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Mathematically, how would air resistance affect trajectory?

Suppose I shoot a cannonball with initial velocity [itex] v_0 [/itex] at an angle of elevation [tex] \theta [/tex]. Air resistance is a force antiparallel to velocity, represented by the equation (according to Barron's Physics C review book!) [itex] \vec F_{air} = - c \cdot \vec v [/itex], where [itex] c [/itex] is a constant SI-expressed in kg/sec.

Because the direction of velocity changes with [itex] t [/itex], so does, as well, the direction of air resistance. To represent its effect in the x & y directions, let [itex] \alpha [/itex] represent the instantaneous angle that [itex] \vec v [/itex] makes with the x-axis. Thus, using 2D Cartesian, I represent the velocities as

[tex] \left\{ \begin{gathered} \vec v_x = \left| {v_0 } \right|\cos \theta - t\left( {c \cdot \vec v} \right)\cos \alpha \hfill \\

\vec v_y = \left| {v_0 } \right|\sin \theta - t\left( {\vec g + c \cdot \vec v \cdot \sin \alpha } \right) \hfill \\ \end{gathered} \right\} [/tex]

*But how do I find y(t) and x(t) ? ?

Suppose I shoot a cannonball with initial velocity [itex] v_0 [/itex] at an angle of elevation [tex] \theta [/tex]. Air resistance is a force antiparallel to velocity, represented by the equation (according to Barron's Physics C review book!) [itex] \vec F_{air} = - c \cdot \vec v [/itex], where [itex] c [/itex] is a constant SI-expressed in kg/sec.

Because the direction of velocity changes with [itex] t [/itex], so does, as well, the direction of air resistance. To represent its effect in the x & y directions, let [itex] \alpha [/itex] represent the instantaneous angle that [itex] \vec v [/itex] makes with the x-axis. Thus, using 2D Cartesian, I represent the velocities as

[tex] \left\{ \begin{gathered} \vec v_x = \left| {v_0 } \right|\cos \theta - t\left( {c \cdot \vec v} \right)\cos \alpha \hfill \\

\vec v_y = \left| {v_0 } \right|\sin \theta - t\left( {\vec g + c \cdot \vec v \cdot \sin \alpha } \right) \hfill \\ \end{gathered} \right\} [/tex]

*But how do I find y(t) and x(t) ? ?

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